Amer. Phy. Soc. interview source |

Sabine Hossenfelder is a physicist at the Frankfurt Institute for Advanced Studies who works on quantum gravity. She is also noted for her BackRe(Action) blog. She has a forthcoming book *Lost in Math: How Beauty Leads Physics Astray*. Its thesis is that the quest for beauty and simplicity in physics has led to untestable theories and diverted attention from concrete engagements with reality.

Today we wonder whether her ideas can be tested, at least by analogy, in computational complexity.

Her book is slated to appear on June 12. We have not seen an advance copy but the book grew from her past commentaries including this from 2016, this in *Nature* in 2017, and this last week. The criticism of string theory goes back even before the book and blog *Not Even Wrong* by Peter Woit of Columbia and the book *The Trouble With Physics* by Lee Smolin emerged in 2006. We are not trying to join that debate but rather to engage with the general thesis she stated here:

Do we actually have evidence that elegance is a good guide to the laws of nature?

She continues: “The brief answer is no, we have no evidence. … Beautiful ideas sometimes work, sometimes they don’t. It’s just that many physicists prefer to recall the beautiful ideas which did work.” For an example, *supersymmetry* is beautiful but has gone far toward a definite “doesn’t work” verdict.

In theoretical computing and mathematics we both remember and preserve beautiful ideas that work. But as bloggers looking to the future as she does, we address ideas that have not yet emerged from shells, to help judge which ones to try hatching. Algorithms and complexity have feet planted not just in Platonic reality but in the empirical fact of programs giving correct answers within the time and other constraints we say they will. Hence we have a testbed for how often *a-priori* beautiful ideas have proved effective and vice-versa.

Certainly the burst of particle physics in the early-mid 20th Century came with unanticipated complexity. We mention one well-known anecdote that, to judge from her index, is not among those in her book: Isidor Rabi won the 1944 Nobel Prize for his discovery of nuclear magnetic resonance, which he used not to treat sports injuries but to discern the magnetic moment and nuclear spin of atoms. When the muon was discovered but appeared to play no role in nuclear interactions, he famously reacted by exclaiming,

Who ordered that?

Muons are ingrained in the physics Standard Model which has much beauty but also has “bolted-on” aspects that those seeking greater beauty seek to supersede. The model is incomplete with regard to gravity and neutrino masses and leaves issues about dark energy and the matter/antimatter imbalance unaddressed.

William of Ockham’s “Razor” is most often quoted as “Entities should not be multiplied beyond what is necessary” in Latin words by John Punch from the early 1600s. Estimating where the bar of “necessary” is set is still an issue. Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred Warmuth in 1987 connected Ockham’s Razor to the complexity of learning, and this was further sharpened by Ming Li, Paul Vitanyi, and John Tromp. Further connections via Kolmogorov complexity and algorithmic probability lead to arguments summarized in a nice survey by Li and Vitanyi with Walter Kirchherr. They quote John von Neumann,

The justification (of a model) is solely and precisely that it is expected to work. … Furthermore, it must satisfy certain aesthetic criteria—that is, in relation to how much it describes, it must be simple.

and continue in their own words:

Of course there are problems with this. Why should a scientist be governed by ‘aesthetic’ criteria? What is meant by ‘simple’? Isn’t such a concept hopelessly subjective?

The answer they seek is that simpler theories have higher probability of having been actuated. This may apply well in large-scale environments such as machine learning and “fieldwork” in biological sciences, in testable ways. Whether it applies on the one scale of one theory for one universe is another matter.

At least we can say that complexity theory proposes grounds for judgment in the physics debate. Hossenfelder seems aware of this, to go by a snippet on page 90 that was highlighted by an early reviewer of her book:

Computational complexity is in principle quantifiable for any theory which can be converted into computer code. We are not computers, however, and therefore computational complexity is not a measure we actually use. The human idea of simplicity is very much based on ease of applicability, which is closely tied to our ability to grasp an idea, hold it in mind, and push it around until a paper falls out.

It hence strikes us as all the more important to reflect on what complexity is like as a *theory*.

We have three main natural families of complexity classes: and and . Up to polynomial equivalence these stand apart and form a short ladder with rungs , then and , then and , and finally which is exemplified among natural computational problems by the computation of Gröbner bases and the equivalence of regular expressions with squaring.

Complexity theory’s first remarkable discovery is that almost all of the many thousands of much-studied computational problems are quantized into the *completeness* levels of these classes. The *reductions* involved are often much finer than their defining criterion of being poly-time or log-space computable. Without question the reductions and quantization into three families are beautiful. Requoting Rabi now:

Who ordered them?

The families intertwine:

The problems they quantize are similarly ordered by reductions. Thus we can extend Rabi with a pun:

Who totally ordered them?

Yet whether these classes are all distinct has escaped proof. The belief they are distinct is founded not on elegance but on myriad person-years of trying to solve these problems.

Stronger separation conjectures such as Unique Games and (S)ETH, however, seem to be hailed as much for explanatory power as for solid evidence. As a cautionary coda to how we have blogged about both, we note that the former’s bounds were shaved in exactly the range of exponential time bounds that the latter hypotheses rely on for their force.

What is also like the situation in physics is a disconnect between (i) how complexity theory is usually defined via asymptotic time and space measures and (ii) concrete real-world feasibility of algorithms, aspects of which we have flagged. This also infects the reliance on unproven assumptions in crypto, which has been remarked by many and may be unavoidable. In crypto, at least, there is vast experience with attempts to break the conditionally-secure systems, a check we don’t see how to have with published algorithms.

Rather than shrink from the physics analogy, we want to test it by going even further with conjectures and comparing their ramifications for theory-building. Here is the first:

Every “reasonable” complexity class is equal to a member of one of the three main families.

Note that some of the big surprises in complexity theory went in the direction of this conjecture. The result that is a perfect example. Also the closure under complement of space shows we only need and do not need its nondeterministic counterpart. Short of specifying exactly which of the several hundred classes in the complexity “Zoo” are “reasonable,” we note that many of its classes are reasonable and such that the equality of to one of the basic time or space classes would be a huge result. For like linear time or space or like exponential time that are not polynomially closed we still get equality to a basic time or space class.

Our second conjecture might be called “superreducibility”:

For every two “natural” computational problems and , either or .

This is roughly entailed by the first conjecture since the three families are totally ordered. It may be viable for finer reductions that collapse complements such as polynomial-time one-query reducibility. It is however false without the “natural” qualifier: whenever but does not reduce back to , there are infinitely many pairwise-incomparable languages between and . We wonder whether one can formulate an opposite of the “gappiness” property used to prove this theorem in order to make the second conjecture more formal.

Combined time-space classes for different pairs may furnish exceptions to both conjectures, but how natural? Eric Allender noted to us that has the first-order theory of the reals with addition as a natural complete problem, as shown by Leonard Berman. It fits between and but equality to either would surprise. It preserves the total order conjecture, however. Closer to home are “intermediate” problems within or in the realm of or or others. We surveyed work by Eric and others that gives some of these problems greater relatability under randomized Turing reductions but less likelihood of hardness. Notwithstanding these issues, we feel it will take a stronger principle to deflate the guidance value of these conjectures.

If we had a choice in building complexity theory, would we build it like this? Should we invest effort to simplify the theory? Is there a model that improves on the Turing machine? Are there theories within computational complexity for which lack of beauty inhibits their development? For one example, Dick and I started a theory of “progressive” algorithms but ran into uglefactions.

The clearest example for our thoughts about theory-building may be Kolmogorov complexity (KC) itself. It is the most direct effort to quantify information. If there is any place where we should expect a simple theory with unique concrete answers and radiant beauty, this is it.

Much as we love and apply the subject, we do not get that tingling feeling. First, the metric by which it is quantified—a universal Turing machine (UTM)—is an arbitrary choice. Any UTM has equal footing in the theory as it stands. The difference made by choice of is just an additive shift related to the size of and the theory is invariant under such shifts. But if you want to know about concrete KC there are strenuous efforts to make.

Second, there are multiple basic definitions, starting with whether the set of code strings needs to be prefix-free. No clear winner has emerged from the original competing proposals.

Third, the metric is *uncomputable*. Proposals for approximating it by feasible KC notions have only multiplied the entities further. One can base them on automata that have computable decision properties but then there are as many notions as automata models. I (Ken) mentioned here a conversation last year among several principals in this line of work that did not radiate satisfaction about concreteness.

Fourth, these issues complicate the notation. Is it or or —or or or —conditioned by default or not on and the like, and are we dropping or keeping additive constants and (log-)logs?

We note a new paper on making the KC theory more “empirical” that may help clean things up. But in the meantime, we cannot deny its importance and success. Our point is that the above marks of ugliness are a brute fact of reality, and any attempt at a more beautiful theory of strings would fly in the face of them.

In what ways might the quest for beauty and simplicity in complexity theory be necessarily compromised? What do you think of our conjectures: like them? refute them?

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*Triangulating proofs to seek a shorter path*

Cropped from 2016 Newsday source |

Mehtaab Sawhney is an undergraduate student at MIT. His work caught my eye on finding his recent paper with David Stoner about permutations that map all three-term arithmetic progressions mod to non-progressions. Here a progression is an ordered triple where . The paper addresses when such permutations can be found in certain small subgroups of while I am interested in senses by which they are succinct. This made me curious about Sawhney’s other work.

Today Ken and I wish to report on Sawhney’s simple new proof of the famous triangle inequality in .

Sawhney presents his new proof in a short note which has just appeared on p218 of last month’s issue of the *College Journal of Mathematics*:

An Unusual Proof of the Triangle Inequality.Summary: A standard proof of triangle inequality requires using Cauchy-Schwarz inequality. The proof here bypasses such tools by instead relying on expectations.

Recall that the triangle inequality for the Euclidean norm on dimensions says that for any vectors and ,

Here as usual the norm of a vector is

The “standard proof” he refers to is represented by this one taken from Wikipedia’s triangle inequality article:

Here Cauchy-Schwarz is used to obtain line 4. Now Cauchy-Schwarz also requires a few lines to prove—indeed one could write a book about it. It feels like the combined proof is tracing two sides and of a triangle, when there ought to be a shorter and direct third side. That is what Sawhney offers.

He can prove the triangle inequality in dimensions by only using the trivial one-dimensional version. That is the fact that for the absolute value

where and are real numbers. Well, it needs the notion of mathematical expected value . This is formally defined via integration on . But he really only needs that expected value obeys some simple properties that one could say are “expected”: additivity, linearity, and ability to manipulate its argument. The one special property is that the norm of a vector in scales as the expected value of its inner product with a unit vector . Formally:

where is a fixed nonzero constant. For one takes the integral of over the circle, which is , and divides it by to make an average, so . The values for higher are different but the difference doesn’t matter, only that is fixed and nonzero. The rest of the proof needs only the 1-dimensional triangle inequality to go from line 1 to line 2:

Pretty neat. No?

I must say that I was quite surprised to see a radically different proof that did not use Cauchy-Schwarz or some equivalent inequality.

Sawhney’s proof is also one a computer theorist could have found. The idea that he relies on is quite neat: the norm of a vector can be computed by taking random projections. This is not elementary but it is intuitive. It is something that “we all know”—yet we did not make the connection to the triangle inequality. This is another example of the power of expectation as a concept.

As Sawhney remarks in his note, the Cauchy-Schwarz inequality can be proved from the triangle inequality by reversing the flow of the proof cited above from Wikipedia, hence it can now be derived via his proof. But that again would be taking two sides of a triangle, while Cauchy-Schwarz has a direct proof. What’s nice is that now both inequalities have a proof that doesn’t reference the other and have a nice bridge between them.

The obvious open problem is: can we use a similar randomness trick to prove other inequalities? Perhaps there are new proofs to be discovered; perhaps there are open inequalities that can be attacked by this method.

[fixed absolute value bars]

Great Discoveries in STEM source |

Claude Bachet de Méziriac was a French mathematician of the early 1600s. He is the first person we know to have posed and solved the problem, given relatively prime (also called coprime) integers and , of finding integers and such that .

Today we revisit some questions about generating coprime pairs deterministically.

Étienne Bézout later observed that for all cases of there are integers such that . The convention of naming this identity for Bézout now extends to the case, so that and are called the *Bézout coefficients* of and . If we count and as being coprime with every integer (including zero) then are coprime if and only if making exist.

Bachet de Méziriac is known mostly for two books. One in 1612 was a compilation whose title translates to *Pleasant and Delectable Problems Fashioned By Numbers* and which inspired subsequent books on mathematical recreations. The other was his translation of the *Arithmetica* of Diophantus from Greek into Latin. It made a marginal contribution to number theory: it contributed the margin in which Pierre Fermat wrote the statement of his famous theorem.

Say we are given a fairly large integer and wish to find coprime pairs near it. It is of course easier to find them by random guessing than to find primes. The primes up to have density only about but the chance of finding a coprime pair approaches about 61%. More exactly it approaches which is the reciprocal of .

The connection extends to other values of the zeta function. Let us take the naturally-extended Bachet condition to be the definition of integers being relatively prime: there exist integers such that . The probability that drawing each from uniformly at random (with replacement) satisfies this condition approaches as .

We can use Bachet’s condition to extend the logic for . We might have no idea what it should mean for one number to be “coprime” but is satisfiable for integer only when and . The probability of drawing from goes to zero, which is consonant with the series for diverging to .

For it is less clear whether to apply the convention that an empty sum is zero. This would be consonant with the literal reading of “” as but not with the analytic continuation . The latter would give as a “probability.” We wonder idly whether Bachet’s condition helps toward a liberalized interpretation that would further give a sensible relation to for fractional real values of and then to complex ones.

But we digress. We want to generate coprime pairs efficiently and deterministically without any guessing.

The PolyMath8 project on “Bounded gaps between primes” generated many interesting sub-projects. Some of them address the issue that if is in the middle of a large gap between primes then simple search up or down will fail. We have blogged about the discovery that searches for primes to use in RSA keys follow the same trajectory to find the same primes so often that simple calls break many keys.

There are various ways to weaken the problem. We can ask to find an integer near that is a prime power. We can ask to find, say, a set of 100 numbers near such that at least 67 of them are prime. Note that we do not allow randomness in the solution to find .

In general we want to improve randomized assertions of the form, “there is probability at least of finding with property ” to “we build a relatively small set of which at least of the members have property .” We can *then* decide whether drawing randomly from or searching through is a better policy, informed by factors such as the relative cost of testing .

We have before talked about the W-trick of number theory. It maps where is the product of all primes below a small threshold and is coprime to (often just ). The inverse image of the primes under is free of biases modulo .

This comment by Ben Green neatly expresses the analytical motivation, as does section 4 of this paper. The freedom from bias simplifies reasoning about the distribution of primes while preserves arithmetic progressions. We are interested in using it and similar tricks to create sets as above—or sets with many coprime pairs—in conjunction with other assumptions.

Here is our first new situation and problem. Suppose that we have two boxes that contain the numbers and secretly. We want to make them into co-prime numbers. We are allowed to map to and map to . But we cannot see the values of and .

This is not easy it seems. We can weaken the goal a bit: *Give a series of translations and so that for most of them the numbers map to co-prime numbers.* Note we do not allow randomness in the solution.

However, suppose that we can compute the sum exactly. Then there is a method for solving this problem that is quite simple:

Pick a prime that is fixed. Then find so that . Add to and add to . This makes them co-prime.

The reason is simple. Note the two numbers are now

Let divide both of these numbers. Since both are positive must be a prime—the original and could have been both zero. Now must divide . This implies that is equal to . And so must have originally divided both and . So we can see now that if we move and to and it is the case that for most in say the numbers are co-prime if we select .

In our second situation, we are implicitly given a pair of numbers for which we cannot determine their values. We can, however, get some partial information about them and can change them to in a certain controlled fashion. This still may not be enough to guarantee that and are coprime.

So we apply our weakened goal: We want to generate a set of pairs such that at least two-thirds of the pairs in are co-prime. We want the method of generating the pairs to be easy to compute and deterministic. We will do this with but hint at what is needed to expand to larger sets .

Definition 1The “2-of-3 model” problem is the following. We assume that we have two natural numbers and , but we have no idea what their values are. We do know the following:

- The value of an even number which is fixed.
- The value of the sum .
- There is so that and a so that .
Finally we can replace by for . The goal is to find

for so that for at least two values of the values

are co-prime. Moreover, we want to be able to find the values in polynomial time.

We’ve chosen some specific constants and terms but the pattern is meant to be generalizable. For the above settings we prove:

Theorem 2There is a polynomial time algorithm that solves the 2-of-3 model problem.

*Proof:* Find a prime so that and is divisible by . This is possible since there are primes in such an arithmetic progression. Now we claim that there is a so that

Moreover we can find this in polynomial time. Note that exists since

which is equivalent to

But is divisible by since divides . Thus exists and is easy to compute. Now let and be so that

Suppose that and have a prime that divides both. Clearly must be odd and thus it must divide and so . But for most splits of we get that this is impossible.

What more can be said about these weaker problems of generating primes and coprime pairs?

The above results are by Dick and this post marks his return after the heart surgery six weeks ago.

The upcoming “Golden STOC” in Los Angeles will be wrapped in a 5-day TheoryFest along lines of last year’s. It will include an inaugural meeting of the TCS Women initiative. Information on travel scholarships to the TheoryFest for female graduate students. We have noticed some recent revived interest in our post a year ago on gender bias, and there is also our theme that CS has seen over its history numerous “Absolute Firsts” for women: “first X” rather than “first woman X.”

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*It makes a fellow proud to be a nerd*

Tasmanian Archive source |

Tom Lehrer is Emeritus Lecturer in Mathematics at Cowell College of the University of Santa Cruz. He is listed not in the Mathematics Department but in Humanities, for which he also lectured on musical theater. He was my first witness that effective input to the social conversation could start from conversancy in mathematics.

Yesterday was his 90th birthday and we hope he had a great one.

We already covered one of his published mathematical papers on statistical modes here. Accordingly it is reasonable for us to divagate into his other output. This consists of some number of satirical songs. A retrospective on Lehrer’s “life of scientific satire” last week in *Nature* put the number at “some 50 songs (or 37 by his own ruthless reckoning)” and the 37 figure was also cited here. Thirty-seven is also the average, median, and mode of plays credited to William Shakespeare, though that number too is trending upward.

Maybe it gets up to 74 songs counting revisions—and frankly repeats—on the tracks of the three–CD set, “The Remains of Tom Lehrer.” That still seems to miss a few of what he termed “some of the old/new math songs” in his remarks before a 13-minute performance at the 80th birthday celebration for Irving Kaplansky, from whom he’d taken two courses. Under ‘new/revised,’ he had added a verse celebrating Andrew Wiles’s proof of Fermat’s Last Theorem (in 1993 as it stood) to a discarded draft song “That’s Mathematics” for a children’s education show that had been dropped:

Andrew Wiles gently smiles

Does his thing, and voilá!

Q.E.D., we agree

And we all shout hurrah!

As he confirms what Fermat

Jotted down in that margin

Which could’ve used some enlargin’.

I knew Lehrer’s songs even before singing “Pollution” in my elementary school’s 5th and 6th grade chorus. My parents played his three main albums all the time. I guess hearing “The Masochism Tango” did not scar me for life. I recognized the political issues. What complemented my captivation was my knowing he came from the world of mathematics.

Indeed, when I memorized and started to perform “The Elements” (*a capella*), I changed the last couplet from

These are the only ones of which the news has come to Hah-vuhd;

There may be many others but they haven’t been discah-vuhd.

to

These were the only ones of which the news had come to Princeton;

And I know that several others have been discovered since then.

This was long before I applied to colleges. The point is how I associated the math/physics legacy. This supplemented the books of Lillian and Hugh Lieber connecting SAM (Science and Mathematics) to society.

Many of Lehrer’s songs were tied to specific events in the 1950s and 1960s, especially those he wrote for the US edition of the BBC satirical TV show *That Was The Week That Was*. His experiences in the US Army (“It Makes a Fellow Proud to Be a Soldier,” 1959) before the Vietnam War fed into his anti-war songs and commentary. This page picks ten favorites and gives YouTube video links.

Several of them transfer to today. For instance, the brilliant “New Math” has been adopted by critics of the “Common Core” movement. The inward-looking criticism of “The Folk Song Army” relates to our current national conversations:

Oh we are the Folk Song Army.

Every one of us _cares_.

We all hate poverty, war, and injustice,

Unlike the rest of you squares.

Most of the new retrospectives (not an oxymoron) are asking a question that’s been asked since my childhood: why did he stop writing? He gave a partial answer in a magazine interview for *People*:

“But things I once thought were funny are scary now. I often feel like a resident of Pompeii who has been asked for some humorous comments on lava.”

That was in **1982**. The quotation is often misdated to 2002. I could rant about our hubris of demanding a better-documented past when we can’t even straighten our information-rich present,

“but I digress.” — T. Lehrer

Instead I’ll pose two questions: who came closest to being Lehrer at other times, and what could have been capable of bringing him back?

Nostalgia is often represented by the line, “where are the snows of yesteryear?” Experiencing the fourth of four nor’easters in three weeks, back in my childhood home near New York City last month, sure answered that one.

The ballad with that line dates to 1461 and is by François Villon. This MathsJam item on the philosopher Jean Buridan quotes it an the end. Here is my irregular translation of the quoted second stanza plus the *envoi*—compare other renditions here:

Where is the great wise Heloise

For whom Pierre Abelard was gelded,

Then with Saint-Denis’s monks was melded?

‘Twas love that brought him to his knees.

In like terms, where would now appear

The queen whose power o’er Buridan

Cast him in the Seine with but a bag on?

Where are the snows of yesteryear?…

My friend, don’t press your enquiries here,

Not for ten years, of where they’ve gone;

Till all your questions leave but one:

Where are the snows of yesteryear?

Can we juxtapose this with Lehrer’s ballad on Alma Mahler Gropius Werfel? There are parallels and differences. Villon entered college young and earned a Master’s, but teaching gave way to an eight-year criminal record. He was no nerd; his rakish lines were backed by experience. He used parody and puns to skewer the proud and high but held to the norms of courtly love. Bitterness and nastiness factored in to create a “mixed tone of tragic sincerity [that] stands in contrast to other poets of the time.”

If our idols say the past was awful more than bright, can we call that “nastalgia”? When the great London impresario Cameron Mackintosh mounted a stage revue of most of Lehrer’s songs, Lehrer’s advice included:

“The nastier the sentiment, the wider the smile.”

The most sage article I’ve found on Lehrer and the limitations of satire is an interview with the Sydney Morning Herald in 2003. It includes Lehrer’s quip from 1973—

“Political satire became obsolete when Henry Kissinger was awarded the Nobel Peace Prize.”

—but moves on to a key difference between *laughing at* jokes and *applauding* them. The latter may be only tribal. “Irreverence is easy, but what is hard is wit.” He could not continue performing more-of-the-same, and for all the greater intensity we say we feel today, more-of-the-same could not bring him back.

So what could? I nominate something that returns to nerd roots in information technology. Here is an example snip from the BuzzFeed story mentioned above as it appeared on my office machine while I was researching yesterday:

source, but YMMV |

You might note the “nastier” quotation and how the paragraph below the ad continues this section’s meditation on satire and humor. But they’re not the point. For that I tell one more story.

Dick continues to recover apace from his heart surgery. Anticipating a visit three weeks ago, I bought a pack of the blandest *nice* cookies I could find: French vanilla kosher macaroons. Not chocolate-dipped, not coconut, nothing to stir the blood. For sundry reasons the visit did not happen (for one, see under ‘Snows: not yesteryear’). So when I got back to Buffalo, I looked online to have a box delivered.

First, good old-fashioned macaroons have been eclipsed in Google searches by *macarons*—as above—an oversweet millennial predilection. I fought for traction until I found a 2014 article, “Where to Buy the Best Macaroons in New York City”—which I still find gets skipped if I put `-macaron` in searches. Further vicissitudes included its being too soon before Passover and online selection limitations. I switched to one place’s vanilla-almond crescent cookies but actual human contact by phone threw up another wall. Finally I chose the ones the article called best—*coconut*—but *Le Pain Quotidien* said they could not deliver unless I went through GrubHub to order. I couldn’t find how to teach GrubHub the inequality (person ordering) (person eating) and my welcome-home message was merged with delivery instructions. But they got there.

My point is, that was three weeks ago. The online mandarins tracked my days of macaroonacy. I am still being pelted by ads for them. Lehrer’s song “Hanukkah in Santa Monica” strangely leaves out Passover, but for me this plague will not pass over.

Surely with our basic *milieu* conspiring so, that needs a skewer. And someone to rhyme those with “spew” and “endu-re.”

Martin Gilbert, the biographer of Winston Churchill whom I knew as a Fellow of Merton College, Oxford, is quoted as naming Lehrer “one of the 10 great figures of the previous 100 years … Many of the causes of which Lehrer sang became, three decades later, part of the main creative impulse of mankind.” Who might rise to match him? Where are the shows of yesteryear?

[a couple word tweaks]

Sports April Fools source |

George Ruth Jr., the “Babe,” may have thought he had cosmic significance but no one knew it until now. He would have said it was all a joke anyway. He certainly loved pranks. As an April Fool’s joke during Florida spring training, he once let it be reported that he had slimmed down to 108 pounds and was beginning a new career as a jockey.

Today we report how the Babe—and every major-league player from David Aardsma and Henry Aaron to Edward Zwilling and Tony Zych—helped uncover a fact about the universe.

This came to light because our correspondents Faadosly and Lofa Polir found a scientific job that suits their talents. They both were hired to the blind injection team at the Laser Interferometer Gravitational-Wave Observatory (LIGO) installation near Livingston, Louisiana.

Blind injection is a protocol whereby a false signal is superposed on the data taken by the main apparatus to test how the main scientific team reacts to it. It is April Fools but with serious intent. As this article noted about the first such trial:

The envelope opened in March 2011 to reveal a fake. The good news was that the team correctly identified the signal. The better news was [that two discrepancies from] what the injection team had expected them to see … turned out to be mistakes by the blind injection team themselves, revealed by the sweat of the [main] LIGO team!

Further successes through 2016 led to implementing the opposite kind of trial where *failure* would count as success. But it succeeded. Let’s describe further what was involved.

LIGO makes inferences about cosmic events from fluctuations on the tiniest of scales, meters, a nano-nanometer. It is subject to random fluctuations of sources from quantum to cosmic. The game is to tell specific events apart from the random background.

Many specific kinds of events are known. Seismic activity is subtracted out by an isolation mechanism that compares absolute and relative motions. Cargo trains that rumble at known times twice daily 7km from the Livingston detector have blunted it enough to count as downtime.

Most local disturbances, however, have been simply identified and discounted by the fact of having two LIGO detectors, the other near Hanford in the state of Washington. Others will open worldwide. True cosmic events will register at both (or all) detectors at precisely known relative times. Signals that show up at only one can be subtracted out.

Of course the *random* events at both LIGOs differ between them. Originally it was not considered terrible to subtract out one detector’s random noise from the other’s, basically because

Yet as an A+ version of LIGO is nearing deployment, it became exigent to test the boundary between systematic and random discrepancies.

The idea was to insert a signal of systematic origin that behaves like random noise—or so we believe. The outputs of strong pseudorandom generators were considered but rejected as artificial. This is when the Polirs suggested a source of hallowed significance in quantum physics.

Stephen Hawking famously conceded his loss of a bet to John Preskill over the black hole information paradox. Preskill’s prize in 2005 was a copy of *Total Baseball: The Ultimate Baseball Encyclopedia*. As the Amazon blurb for the current edition states:

About half of the volume is made up of detailed statistics for every player ever to appear in a major league game. Other statistical sections, including records, awards, and MVP and Hall of Fame voting results, help round out this tribute to the statistical minutiae that fascinates many baseball fans.

The edition exists in digital form. As a tribute to Hawking, the blind-injection team agreed to use this as the signal.

Some initial processing was done on the data. Symbolic categories like hit, homer, strikeout, and walk were converted to digital form. Obvious redundancies such as each event’s contribution to pitching and hitting stats were subtracted out. Known biases such as Benford’s Law were removed by non-lossy transforms. Not just the 2,300 dense pages of statistics in the print edition, but also the acres of raw recorded in-game data from which they were compiled in later seasons, were refined into a 6.4 gigabytes stream that was believed indistinguishable from white noise.

The final key was that what they fed as to Livingston was not , but rather the item-scale difference—essentially the *XOR*—of with the signal detected milliseconds earlier at Hanford. This was done on March 17. The blind team had control of the system clock so that the scientific teams attached to both places would not know of the delay. Hence the teams’ own differencing would give not *random* but rather back again.

It should be understood that much of the “sweat” quoted above came not from the human team members but from supercomputing of incredibly high bandwidth using a sever farm in Quincy to the north of Hanford. The new massively parallel deep neural rules-based classification algorithm by Xiaowei Gu et al. was used to build *generative models* for all of the Livingston stream , the Hanford stream , and their item-scale difference which was . This was the shock:

The generative model for filled only 300 megabytes.

Not only that, the model was stratified so that, for example, the records for 2017 were compressed under 4.5MB. The upshot is:

There is a file smaller than a medium-quality JPEG photo from which the entire recorded events of the 2017 Major League season can be recovered verbatim with shallow post-processing (mainly applying the model rules that were found and then undoing the transforms mentioned above).

We can compare this with the stunning main result of the recent AlphaZero paper as noted in comments to our post on it and summarized here:

There is a file smaller than 300MB such that relatively shallow processing with access to trounces the deep search of the world’s best chess programs, with results that appear to border on perfect play.

To focus our comparison, note that perfect play is known for all positions with 7 or fewer pieces in files that fill over 100 terabytes at the Lomonosov Moscow State University computer center. Those files can be generated from a very tiny file , one that merely specifies the rules of chess and the allowed contents of the board. However, the file cannot be efficiently consulted—the very creation of from characterizes the notion of computational depth. The shock of AlphaZero is that *the entire game of chess* has been compressed into a small file that is also consultable.

What doubles the shock in baseball is that unlike with chess its *outcomes* are not rule-based. Once the batter hits the ball much of what happens is ascribed to luck. Per discussion in our previous post, no convincing evidence of a “hot hand” in baseball is extant, nothing to distinguish results of at-bats from rolling dice.

Yet the outcomes in the tables of each game over the entire season were revealed as the consequents of relatively simple rules. It is as if some cosmic power decreed:

The first pitch of the season shall be a home run. A light-hitting infielder shall hit home runs as the only scores in back-to-back 1-0 wins.

By the correspondence of information complexity to entropy, there is less entropy on specifying the first pitch than the 107,348-th as yielding a homer, or in making a team’s second game a Xerox of its first. This analogy is imperfect—of course homers will be hit on other pitches and there were other differences in the Giants’ two 1-0 wins. But it is enough to illustrate how rules can be simpler than listing random events. The point is that like the particular neural weights of AlphaZero after its training, the values and implied rules may be subtle, unexplainable, and felt only by their effects. But they are present and they are *short*.

Both AlphaZero’s success and the new discovery about baseball rest on the great broken syllogism of physics that flows from Hawking’s concession:

- Information can be neither created nor destroyed.
- The universe began in a state with initial conditions of low complexity.
- We currently observe high complexity pumped up by quantum noise.
- ???

The likeliest resolution may pin the discrepancy on a local/global difference. The Mandelbrot set is specified on the whole by a tiny equation, but individual vantage points on it can have high complexity. Quantum outcomes have with overwhelming likelihood positioned our local world at such a point.

Yet the efficiency of randomness generation by this process need not be 1:1. The new results from the LIGO supercomputers on the entropy of baseball suggest an upper bound of 1:20 or so. It is as if every Babe Ruth we see is really slimmed down to a jockey’s weight.

In all seriousness, how much entropy could we extract from the statistics of a baseball season?

We are grateful to the Polirs and their superiors for permission to report on these developments in advance of normal scientific review.

]]>

*And if so, what units does it have?*

Cropped from source |

Fabiano Caruana has just won the 2018 chess Candidates Tournament. This earns him the right to challenge Magnus Carlsen for the world championship next November in London. A devastating loss on Saturday to the previous challenger, Sergey Karjakin, had seemed to presage a repeat of his last-round failure in the 2016 Candidates. But Caruana reversed the mojo completely with powerful victories in Monday’s and Tuesday’s last two rounds to win the tournament by a full point.

Today we congratulate Caruana on his triumph and hail the first time an American will challenge for the world championship since Bobby Fischer in 1972. And we ask, is there really such a thing as being “in” or “out of” *form* in chess and similar pursuits?

Summer 2014 saw Caruana pull a stunning streak that put him in a pantheon with Fischer and the past champions Mikhail Tal and José Capablanca. Following a stunning 1.5 point margin of victory in the elite Dortmund Sparkassen event in only 7 rounds, he rattled off 7 straight wins in what remains the highest-rated tournament in history, the 2014 Sinquefield Cup in St. Louis. The streak included a win over Carlsen, and he had Carlsen dead-to-rights again before allowing a draw which stopped it. His performance rating for the fortnight was over 3100, high above Carlsen’s regular rating of 2877 and his own 2801, while I measured the intrinsic quality of Caruana’s moves at “only” 2995 during the win streak and 2925 overall.

Caruana had had a head of steam coming in to the 2016 Candidates from a strong tied-2nd behind Carlsen in the January 2016 Tata Steel annual classic in Wijk aan Zee, Netherlands. In the 2016 Candidates, he had been tied for the lead before losing to Karjakin in the last round. By contrast his run-up to the 2018 Candidates was listless. A roundup of predictions at Dennis Monokroussos’s blog *The Chess Mind* included the opinion by a commenter:

“…Caruana seems to be off form, given his rather dismal performance at Wijk aan Zee.”

But what is *“form”* anyway? Does it exist? That is our subject.

Players and teams have been said to “have momentum” for decades. Belief that human competitors innately run “hot” and “cold” is endemic. Only in recent decades have there been attempts to measure such attributes or tell whether they exist at all.

We have covered the “hot hand” question in basketball and there was further controversy a year ago. The arguments turn on whether the distributions of performance outcomes are statistically indistinguishable from distributions with identifiably ‘random’ causes. A fair coin cannot “be hot”—and dice players say “the dice were hot” only in retrospect of streaks that enabled them to win.

However, being *off* form is certainly real when one is ill or otherwise distracted. Is there an obverse? Can one be “super well” (leaving aside performance-enhancing drugs)? Is being “in the groove” prompted by physiological conditions, ones that self-reinforce?

This leads into a further matter that reinforces the “it’s all random” interpretation but also feeds into my alternative view. Suppose being under-the-weather or concretely burdened happens once every five tournaments in a way that drops your performance by 200 rating points. It follows that in the other four tournaments you’ve averaged playing 50 points higher than your published rating. By the Elo expectation table, the extra 50 points gives 57% expectation against equal-rated opponents. This translates to an extra draw or win over a 9-round tournament and is enough of a difference to be felt as “being on.”

Thus a finding that players were “in form” more than random simulations might expect could be explained by this, if the simulations naturally took each player’s published rating as the baseline for their projections. This extends the reach of outcomes to judge consistent with the random-effect hypothesis. Studies of game results from massive data could test this hypothesis. I have not expressly done so with my full statistical model, and this comment in the last basketball item above asked whether it has been done in chess. I can, however, give a partial indication that tends toward “no hot hand” in chess.

The Candidates Tournament was a traditional double-round-robin with eight players and fourteen rounds. Chess tournaments using the Swiss System can have hundreds of players. Their open character and often-high prize funds make them most exigent to screen for possible cheating. The top 10 or 20 or more boards are usually played with equipment that automatically records the moves, which can be broadcast live or with some minutes’ delay. All remaining games are preserved only on paper scoresheets or handheld devices (from) which both players are required to submit. Some tournament staffs painstakingly type the paper games as well into PGN files but others do not. The latter I distinguish by putting “Avail” into their filenames.

In the first round the players on the top boards are those with the highest ratings and their opponents are from the third or second quartile according to the pairing system used. There is no selection bias in those top games. But in all succeeding rounds, the top-board players are those who have kept or earned their place by winning. The “Avail” files hence should be biased toward players who are “in form.”

My screening tests use simple counting metrics: the number of agreements with the chess-playing program(s) used for the tests and the total error judged by the computer in cases of disagreement. The latter is averaged over all moves, agreeing or not. The games are analyzed in the programs’ quick “Single-PV” mode, which is also their playing mode, rather than the “Multi-PV” analysis mode used by my full model which takes hours per processor core per game. If the screening test raises any concern, then the full model can be run for the particular games involved.

The screening test likewise does not support the computation of an “Intrinsic Performance Rating” (IPR) for the games, as I did for Marcel Duchamp and George Koltanowski recently using my full model. But the data is large enough—and the metrics concrete enough—that if there were an “in form” effect then I would expect to see distinctly higher values in the “Avail” files.

I do not. The values I get from the two strands of Open tournaments agree well within the error bars for several hundred tournaments of each kind per year. The “Avail” figure is a little bit *lower* each year since 2016, per each of the Stockfish and Komodo chess programs. The two averages of the average ratings in the two kinds of tournaments, the latter averages weighted by moves in games, are close.

This is only from a quick test, giving more an “absence of evidence” verdict than “evidence of absence.” But it speaks most particularly against the hypothesis that form carries from one *game* into the next day, so that it would be denominated in units of *tournaments*. Whether it might carry from *move* to *move*, so that the unit becomes “playing a good *game*,” may require my full model owing to covariances between moves that only it can compensate for. We can, however, consider a different notion of “form” that brings human qualities inside the ‘random’ picture.

The famous baseball manager Earl Weaver once defused a question of whether his team had momentum by retorting:

“Momentum? Momentum is the next day’s starting pitcher.”

What he meant includes all of these: His hitters who were hot could be stopped by an excellent opposing pitcher. His hitters who had struck out were accustomed to shrugging off one bad day and being fresh for the next, when the pitches might be easier to hit. His own team’s fortunes would most likely depend on his own starting pitcher who hadn’t pitched in four or five days.

In chess, however, losses are said to “stay with” players. They are rarer for the elite and harder to slough off. That’s what made Caruana’s two-win finish remarkable in chess. It was under the same last-round pressure that had accompanied his defeat in 2016 and that had tangibly caused the shocking double–loss finish to the 2013 Candidates. His round-12 loss last Saturday was horrific, a “positional crush.” Not only did he fall into a tie with Karjakin after having led alone since round 7, he stood to lose any tiebreaker with Karjakin or the player next on their heels, Shakhriyar Mamedyarov. So he not only needed to right the ship, he needed to rev it.

If I were to simulate tournaments randomly based on these eight players’ ratings, I would expect to generate a fair share in which the winner finished loss-win-win. Maybe there were random factors that pinged in Caruana’s brain at the right times—not to mention going “pong” in the brains of his two unfortunate victims. My model can tell “ping” from “pong” modulo high error bars. Here are its results for the last three games—with the caveat that they come from a “version 2.5” that has been fitted but not fully vetted and which is a way-station while issues in my intended “version 3.0” are still churning.

- Karjakin 2800 +- 675, 1-0 in 48 moves over Caruana 2760 +- 450.
- Caruana 2585 +- 825, 1-0 in 39 moves over Levon Aronian 1740 +- 1170.
- Alexander Grischuk 2425 +- 535, 0-1 in 69 moves to Caruana 2575 +- 485.

Since the players were all rated near 2800 this suggests more “pong” than “ping,” notwithstanding the enormous two-sigma error bars for measuring just one game. However, the last game’s figures arguably mislead because Caruana after turn 40 had Grischuk in a mortal lock and took his time—as observed here and here—rather than rush in with quick kills seen by the programs, while Grischuk had to try to thrash about. If we cut off at turn 40 then the figures become:

- Grischuk 2755 +- 745, 0-1 to Caruana 2940 +- 540.

This has its share of “ping.” For the whole tournament, my model tabs Caruana right at 2800, with less-yawning error bars +- 160, and the level of his opponents’ moves collectively at 2640 +- 205.

I regard the error bars in my model as reflecting an innate lower bound on human variability. “Pings” and “pongs” happen in a way analogous to quantum uncertainty. But my final point is, they are *human* pings and pongs, not those of coins or dice. The training of my model at Caruana’s level is based on the actual recorded performance of his human peers (no computer games) over the history of chess. Those pings are in the brain, and even in basketball the pings outside the brain are in well-conditioned muscles and smooth joints.

Thus the human factors need to be inside, not outside, the randomized models—inside where our humanity retains the credit for them. In the sporting terms that chess professionals recognize, Caruana showed true grit coming back. He showed the most consistent command from his round-1 win over fellow American Wesley So clear to the end. He was not sick and he put distractions aside, including his loss to Karjakin. Maybe we say this only in retrospect, and maybe the negative results on momentum say that the next time the situation comes up after round 12 we should not bet heavily either on roaring comeback or collapse. But the victory and the strong play on the whole, coming from his brain, constitute his having been *in form*.

How should questions of “momentum” and “form” be formulated—and should the two be treated differently?

Answer to the leprechaun puzzle in the previous post: N = 1 is the count of times Neil’s words use that letter, and similarly P = 0. Those are the two sufficient arithmetical conditions for P=NP, which is Neil’s theorem of choice. (Cued by hints in the text including Neil’s nerdiness.)

[minor word and format tweaks]

Neil L. has graced these pages many times before. Every eve of St. Patrick’s Day he has visited Dick. Sometimes Dick has been hard to find, but Neil has always managed.

Today we relate some medical news while wishing a Happy St. Patrick’s Day 2018.

Neil knew to go to the apartment shared with Kathryn which is adjacent to MoMA in Manhattan. He appeared and took in the grand view of St. Patrick’s Cathedral through their east-facing window. But he did not find either of them there.

What Neil found was a sheet of paper on a table near the window. “A message for me?”, he breathed. He read in big letters across the top “IDEA FOR GOLD-” and that was enough. He snatched the sheet and vanished. But he had to go somewhere. So he came to me.

I had seen Neil for the first time only a year ago. Since I knew what Neil didn’t, I was expecting him. I had gone down to my basement ostensibly to watch “March Madness” while on the exercise machine. I was riveted by bottom-seed UMBC sinking three straight 3-point shots to hold #1 Virginia to a 21-21 tie at halftime, but at the first sign of green smoke I switched off the TV and pulled up two chairs and a small table with ashtray.

Neil intoned as he lit his pipe,

“A blessed eve to ye.”

I replied, “Same to you—I guess it has already turned St. Patrick’s Day in your home isle.” Neil nodded and as he was about to speak I interjected, “You did not find Dick—”

“Aye—aught I saw of him.”

I had permission to inform Neil of why:”He is in intensive care after heart surgery. Kathryn is with him.”

Neil doffed his green hat with a long and serious “Ahhhh…” Then he took a long drag on his pipe. “To be mortal…”, he whispered. But forcing his lip corners brightly up, he said,

“Yet ideas are immortal—that is why I come. Every year, at this time. I have a message from Dick to show ye.”

He pulled out the sheet. I read the entire top line: *Idea for Goldbach*. “He means the Goldbach Conjecture,” I informed Neil. I expected Neil to recognize the conjecture—if leprechauns can be nerds, he is one. Duly Neil intoned:

“Every *x* > 2 that is divisible by 2 is the sum of two—“

“Primes.” I completed his sentence—the pause struck me as strange—and I went on: “With Fermat’s Last Theorem having been solved, Goldbach is now the easiest unsolved problem in mathematics to state. The fact that Pierre Fermat got 357 years of credit for a ‘Theorem’ just because he left a marginal note saying he’d proved it emboldened Godfrey Hardy…” On my new I-Pad I called up the story:

Hardy was known for his eccentricities. … He always played an amusing game of trying to fool God (which is also rather strange since he claimed all his life not to believe in God). For example, during a sea trip to Denmark he sent back a postcard claiming that he had proved the Riemann hypothesis. He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his “last theorem.”

A long green puff accompanied the reply,

“Would Dick do that?”

I reflexively replied “naw” with my mouth but the part of me that really wanted to see the idea had control of my hands. I reached for the sheet and there was a flash of white light.

The sheet no longer had Dick’s handwriting. I expected Neil to chide my haste and withdraw it, but instead he laid it flat and folded his arms:

src |

I goggled a bit, but I knew what I was looking at without googling. “That’s not leprechaun writing. That’s Nigerian script as used in the movie *Black Panther*.”

“Aye.”

“You can’t do that. That’s cultural appropriation.”

Neil emptied his pipe, straightened his gold buckle, tipped his hat again, folded his arms the other way, and gave me a long stare.

“Oh who would ye tattle me to, the First Minister o’ the Irish?”

Neil meant Leo Varadkar, but why “*first*” minister? This was the second time he had avoided saying “prime.” Nor did he use the proper title *Taoiseach* or say “Ireland.” Clearly he was trying to convey something beyond a lesson of intercultural embrace. Neil piped up again:

“It matters aught what the characters *are*, but which ye tell away from each other.”

Neil was right—the information content resides wholly in the ability to distinguish pairs of symbols. Impatiently I asked, “So what is the information? Can you read it?”

“It is coded with the starkest cosmic scrambler—a black hole—so that I may tell little from it.”

My thoughts sprang ahead owing to this week’s passing of Stephen Hawking. In a children’s book written with his daughter Lucy Hawking and Christophe Galfard, the children’s astronaut father falls into a black hole but is resurrected through the Hawking radiation, though it takes a galactic computer “a long time” to reconstruct him. On the serious side, Dick and I have wanted to blog about the proposed solution by Patrick Hayden and Daniel Harlow to the black hole firewall paradox, whereby the one-wayness of Hawking radiation staves off the reckoning of the paradox. I’ve also wondered whether every computational process that *can* occur in nature *must* occur in proximity to any black hole, combined with every computation we can program *ipso facto* being one that occurs in nature. I then sprang back, however, to what Neil had said about immortality, so I asked:

“Neil, from your immortal perspective, would you say that Hawking of all humans came the closest to states that you recognize as pertaining to immortal beings?”

Neil unexpectedly flinched from my question.

“Steve—“

“*Stephen*,” I corrected. “It makes a difference.”

“—he had little to share with us folk. Try Google. You will see.”

Indeed, there was basically no webpage connecting “Hawking” to “leprechauns” at all. Plenty for trolls and a few for elves, plus “fairy” as an adjective, but none for leprechauns. Nor gnomes. Noting what Hawking said about the brain possibly continuing in articles like this and this, and that he also had his DNA sequence shot into space on the Immortality Drive, I pressed Neil on my question. After long pause he replied softly:

“What do ye most celebrate about him this week, after all? If ye look at the Web everywhere it seems…”

Indeed I have been struck how so many of the memorial appreciations of his life were playing up his *human* qualities. The stories… Well, Hawking was human after all.

“Aught may ye have both ways, me lad.”

Whatever Neil meant by “aught,” there was the sheet of Dick’s ideas to decode. Neil had said he might tell a little from it. Jumping from *Theory of Everything* to the technical pivot of the *Imitation Game* movie, I realized we did have some of the plaintext: ‘idea’ and ‘Goldbach’ and associated words. Neil understood and gave it a go.

After much play of green light over the sheet, Neil sighed and announced:

“I could decode just this early theorem.”

Neil copied it out in his hand and I read:

Theorem 1Suppose that almost all even numbers are sums of two distinct primes. Then almost every prime is the middle of an arithmetic progression of length three.

*Proof:* Let be given. We can clearly assume that all even numbers larger than are the sums of two different primes. Let be a prime and let where and both and are primes. Take

Now , , and form a progression of length three. But each is a prime:

and

So this leans on the slight strengthening of Goldbach where writing as is disallowed. After every even number tested has been written as the sum of two different primes. An equivalent form is whether every number is the *average* of two distinct primes. Dick’s conclusion is just the restriction where itself is prime.

Is this open? It still is. Neil and I scoured the writing but we could glean nothing more. Even to prove the existence of infinitely many length-3 progressions of primes had been difficult in the 1930s. What else was there about Goldbach, or did Dick’s sheet move on? I am looking forward to asking Dick next week when he will be up to having visitors.

Neil looked at his watch and gave a start.

“Begorrah—I must be off. Yea though the Irish laddies missed the basketball this year, still I must take care of malarkey me fellows might wreak… To boot, the ladies start tomorrow.”

I had time just to ask one more question: “Neil, if you were in this position and wanted to make sure your ideas for a big theorem were put down—even if not sure which side is true—which one would you choose?” Neil replied:

“With the totality of my uttered words here I have told ye. They have held throughout two attributes each of which literally makes it follow.”

And with a green flash he was gone. I turned on the TV and saw instantly that he was too late: “UMBC 74, Virginia 54” flashed on the screen.

Can you process Neil’s speech to find his answer?

What chance might there be of proving that every prime is an average of two other primes, short of proving the full Goldbach Conjecture?

I am sure all our readers will wish Dick a safe and speedy recovery. We were working on this on Thursday before his operation, including the math.

**Update (3/18)**: Dick continues mending in the ICU. As for Neil L., he either failed to contain leprechaun excesses or joined them. This is what a graph of leprechaun involvement looks like.

**Update (3/27)**: Dick is now home from the hospital and is resuming normal activities.

**Update (3/28)**: Good news continues; puzzle answer at the end here.

Cropped from “Knuth at Brown” video source |

Donald Knuth’s 80th=0x50th birthday was on January 10. In the array of his birthdays, numbering from zero so that stands for his birth day in 1938, that was indeed . However, as the 81st entry in the array it might have to be called his 81st birthday. Oh well.

Today we salute his 80th year—wait, it really is his 81st year—and wish him many more.

Our little riff on “off-by-one” issues is not an idle matter. Don’s epochal multi-volume monograph *The Art of Computer Programming* (TAOCP) set standards for presenting as well as designing algorithms and programs. He nodded to community agreement on the benefit of “numbering from zero” but began his chapter on lists and arrays by numbering from 1 before using either convention. The xkcd cartoon “163: Donald Knuth” projects onto him the opinion,

“Different tasks call for different conventions.”

In his textbook *Concrete Mathematics* with Ron Graham and Oren Patashnik, in a passage indexed as “Zero not considered harmful,” they say:

People are often tempted to write instead of because the terms for , , and in this sum are zero. … But such temptations should be resisted; efficiency of computation is not the same as efficiency of understanding! …[S]ums can be manipulated much more easily when the bounds are simple. … Zero-valued terms cause no harm, and they often save a lot of trouble.

Emphasizing zero *values* rather than the index , they continued with what they termed “a radical departure from tradition”:

Kenneth Iverson introduced a wonderful idea in his programming language APL … to enclose a true-or-false statement in brackets, and to say that the result is if the statement is true, if the statement is false … This makes it easy to manipulate the index of summation, because we don’t have to fuss with boundary conditions.”

This was almost 25 years ago but Don’s advice is still a step ahead today.

I (Ken) used to think that only chess held marquee events up near the Arctic Circle: the 1972 match between Bobby Fischer and Boris Spassy in Reykjavík, Iceland; the 2014 Chess Olympiad in Tromsø, Norway. The January 8–10 workshop and celebration for Don’s 80th birthday was organized in Piteå, Sweden, which is just north of 65° latitude.

“Organized” is the operative word. As Don says in the opening seconds of a video with the science editor of Sweden’s premier newspaper *Dagens Nyheter*:

It took such a perfect match. I don’t believe that any other … anywhere else in the world would come anywhere near being right…

We have elided one of Don’s words, and we’ll keep you in suspense about it, but for a hint it’s the kind of suspension. Which is one of the more difficult things I’ve ever had to do in plain TeX, because WordPress does not recognize fancy add-ons to LaTeX (nor even the \TeX or \LaTeX macros).

TeX, of course, was Don’s free gift to the world. It sprang not only from his desire to make mathematical typesetting freely available—and to demonstrate how to code large-scale useful software—but also from the value of pliable visual beauty. It was duly featured at the workshop in a talk by by Yannis Haralambous titled “TeX as a Path.” Here are the other talks in the order they were given—most have slides on the talks page:

- Robert Sedgewick: “Cardinality Estimation”
- Michael Drmota: “Subgraph counting in series-parallel graphs and infinite dimensional systems of functional equations”
- Tim Roughgarden: “Don, Stable Matchings, and Matroids”
- Svante Janson: “Random permutations avoiding some patterns”

Sunset was observed during the lunch break—this was the Arctic Circle in early January, after all.

- Wojtek Szpankowski: “Analytic Information Theory: From Shannon to Knuth and Back”
- Richard Stanley: “Combinatorics and Smith normal form”
- Ronald Graham: “Eulerian Adventures with Don”
- Richard Karp: “The Rectilinear Group Steiner Problem”
- Sunrise
- Susan Holmes: “Don Knuth and the Reproducible Research movement”
- Jeffrey Shallit: “Using Automata to Prove Theorems in Additive Number Theory”
- Anders Björner: “To where Knuth paths can lead”
- Sunset
- Persi Diaconis: “Don Knuth and Backtracking”
- Svante Linusson: “Using a modified Robinson-Schenstedt-Knuth correspondence on randomized sorting networks”
- Michael Paterson: “An old buffers problem: taking a closer look”
- László Lovász: “Graphings, hyperfiniteness and combinatorial optimization”
- Robert Tarjan: “Zip Trees”
- Gregory Tucker: “Tales of Computational Geomorphology”
- Sunrise
- Martin Ruckert: “Programming as an Art”
- Erik Demaine: “Fun and Games meet Computer Science”

There followed the talk on TeX by Haralambous, a tribute by Don’s son John Knuth, and a brief introduction by Jan Overduin to the signature event of the day, which took place after the cutting and serving of the birthday cake.

The word *apocalypse* comes from Greek *apo-* “away” and *kalupsis* “covering”—that is, a revelation. As the original Greek name for the Book of Revelation it acquired its connotations of catastrophe and final destruction. What we now consider to be apocalyptic writing goes back at least to the fall and Babylonian exile of Israel and Judah. But the primary element of revealing *hope* sets Revelation apart—except that its germ is in the last chapter of the book of Daniel.

As Björner highlighted in his talk, in a January 1981 interview that was published a year later in the *The Two-Year College Mathematics Journal*, a caption opened by noting him as “an accomplished organist and composer” and went on to quote him:

“I want to write some music for organ with computer help. If I live long enough, I would like to write a rather long work that would be based on the book of Revelation. The musical themes would correspond to the symbolism in the book of Revelation.”

He got it going in early 2011. So January 10 saw the world premiere of his *Fantasia Apocalyptica* on the Orgel Acusticum, whose official page begins by saying, “This is an instrument for the 21st century.” It was built in 2012 by Gerald Woehl. A year ago, Don visited it in Piteå and wrote an incredibly detailed exegesis, including the organ’s software features and controls.

You can hear parts of it in an introductory video by the Canadian organist Jan Overduin, who performed it in Piteå and will do so again on November 4 at his home First United Church in Waterloo. This video is atop Don’s own page which includes his full score in manuscript and typeset forms.

Both Overduin and Don describe how the music closely follows both the message and the numerical contents of the Book of Revelation. Don described the process as “Constraint-based Composition” in a lecture of that title in May 2015 at Stanford. He draws analogy to constrained writing as practiced by the Oulipo group of mainly-French writers. A very loose example is how every post on this blog is constrained to follow some “GLL invariants.”

However, what I (Ken) think of as the highest example of constrained writing is *translation*. Insofar as they give scope for the translator’s own creativity, they are constrained by faithfulness to the source text. Creative choices come because meaning and imagery and emotion require different mixings in different languages and media. The *Fantasia Apocalyptica* is organized into one movement for each chapter of Revelation, and each movement follows the verses as Overduin expounds. It is thus a musical translation at perhaps a finer grain than many tone poems that have been based on literary works.

With “fine-grained” as well as “constraints” we have circled around to computer-science concepts again. If there is one over-arching point we see Don conveying, it is that such integration of informatics with arts and language and real-life appreciation should be natural.

We wish Don many more birthdays to come, no matter how they are numbered.

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*How might it be applied in complexity theory?*

St. Andrews history source |

William Burnside was a well-known researcher into the early theory of finite groups.

Today Ken and I thought we would talk about one of his most-cited results—a result that is really due to others.

This happens all the time in mathematics. In this case Burnside wrote an important book on finite group theory and included a lemma that is called various things. It is sometimes named for Augustin-Louis Cauchy or Ferdinand Georg Frobenius, or called, “the Lemma that is not Burnside’s.” The lemma was incorrectly attributed to Burnside because he proved it in his 1897 book *Theory of Groups of Finite Order*. His title for the last two sections of his eighth chapter was the statement of the lemma:

Number of symbols left unchanged by all the substitutions of a group is the product of the order of the group and the number of the sets in which the symbols are interchanged transitively.

After his proof in section 118, he began section 119 by writing:

119. The formula just obtained is the first of a series of similar formulae, due to Herr Frobenius,* which are capable of many useful applications.

The * citation was to a one-page paper by Frobenius in *Crelle’s Journal* in 1887. The formula itself appeared in an 1845 paper by Cauchy. In his 1911 second edition, Burnside stated it in section 145 (page 191) as “Theorem VII” where he adjoined a twin statement about summing the squares of the numbers of fixed elements.

The book has no instance of the word “lemma,” which the first edition used only in section 77. There is no nearby mention of Cauchy or Frobenius and neither the 1887 nor the 1845 paper is cited anywhere. Instead, Burnside’s next mention of Frobenius comes on page 269 astride five new chapters on representation theory that were his great thrust in the expanded edition. He used a big * footnote stretching across two pages to cite multiple works by Frobenius *en banc*:

* The theory of the representation of a group of finite order as a group of linear substitutions was largely, and the allied theory of group-characteristics was entirely, originated by Prof. Frobenius. …

That Burnside gave a tandem statement supports the position that he felt it was all well-known research, but perhaps what the online *Encyclopedia of Mathematics* calls the “mysterious dropping” of Frobenius owed as much to his sweep toward this grand encomium.

Burnside is more properly known for actual beautiful results such as the important theorem, which shows that every finite group whose order has only two prime divisors is solvable. The original proof relied heavily on representation theory, and it took many years to get a proof that avoided needing this machinery—see this for the theorem and a short historical note.

Then there is Burnside’s conjecture, which lasted for six-plus decades before being refuted in 1964, but which still has viable forms and impacts as we covered before.

But what of the Lemma? The problem was less Burnside’s lack of citation and more the habit of others citing it from Burnside. So we propose calling it the Lemma Cited From Burnside (LCFB). The initials can also stand for “Lemma of Cauchy-Frobenius per Burnside.”

An *action* of a group on a set is a mapping from to permutations of that multiplicative: . That is, the action is a subgroup of the symmetric group of that is a homomorphic image of , but the action retains information about which homomorphism was used.

An action induces—and is equivalently definable as—a function

with the property that for all and , . The *orbit* of under the action is the set of values over all .

This is all made more visual if we write for . Especially when the action is *faithful*, meaning is 1-to-1, we can picture the elements of as mapping -es directly. Then the orbit is . Some questions to ask are:

- How many elements are fixed by a given , namely ?
- How many elements are fixed by the action on average?
- How many orbits does the action have?

The latter two questions may seem unrelated. But the LCFB says that *their answers are the same*:

Lemma 1Let act on the set . Then

The key idea of the proof is that orbits are balanced. For any , define its *stabilizer* by and note that this forms a subgroup of . The *orbit-stabilizer theorem* states:

Theorem 2Let act on the set and let . Then

It follows right away that for all in the orbit , ; call this . The proof of Lemma 1 comes quickly too:

which is what we needed.

We have used a “Theorem” to prove a “Lemma.” Is the “Theorem” obvious? We note substantial discussion about that in a StackExchange post and links from it to a lengthy post by Tim Gowers. We will try our own explanation because it speaks right to aspects of the cycle structure of permutations that we are trying to quantify in new ways.

List out the orbit of as . For each , pick an element of such that . And list out as . We claim that

If they are equal as elements of then , but by the choice of and this forces . And . Thus the elements are all different.

Now consider any . There is a such that . Now put . Then

Therefore belongs to ,so it equals for some . Thus , so we have written . Thus the elements run through exactly once. It follows further that for each there are exactly elements such that , namely those for different . It follows that , which proves the theorem.

Note that we seem to have avoided appealing to the concepts of *order* (of an element or group) or *quotient* that go into statements of Lagrange’s theorem. Of course they are present, but we used as a screen to hide them. We didn’t even have to exhibit a 1-1 correspondence between and . It all simply flows from the axiom of inverses and multiplicativeness of the action which in particular gave us . There is no intermediate notion of how a group can act on a set—it must have perfect balance in orbits.

Here is an example that also shows the difference between the physical and algebraic nature of an action. Consider regular -gons whose edges are colored one of colors. Equivalently, we can consider them to have triangular facets colored the same front and back. The physical actions are rotating the polygon right by degrees and flipping it over. These generate the *dihedral group* . The number of orbits of the action of on the set of tile colorings tells us how many tile types there are.

The LCFB tells us to count the colorings fixed by each group element. Consider and , that is, black and red colorings of a square. This gives permutations and colorings. We count as follows:

- The identity fixes all 16 colorings.
- The 90-degrees-right () and 90-degrees-left () rotations fix only the two monochrome tiles.
- The 180-degree rotation fixes those two plus the two with diagonals of opposite colors.
- The flip shown at top below fixes all tiles that have facets 2 and 4 the same color, 8 in all, including the coloring shown.
- The flip plus 180-degree rotation fixes when 1 and 3 have the same color, giving another 8.
- The flip plus rotation fixes the 4 colorings with 1 and 2 the same color and 3 and 4 the same color. The flip plus does likewise for 1 and 4 vis-à-vis 2 and 3.

We get fixed items, hence the LCFB gives orbits.

Two different representations of a flip element in . |

The bottom part of our figure shows how we could have oriented the square like a diamond before doing the mirror-image flip. Now, however, the coloring shown is not fixed, and only 4 not 8 colorings are preserved. Is this an inconsistency? Although the flip is physically the same, it is algebraically different. Represented as a permutation in cycle form, the original flip was , whereas the latter flip is . The latter flip is equal to following the former flip with the rotation, which indeed fixed 4 colorings under the former representation of the action of .

This suggests that the *cycle structure* of the permutations used in the action is what matters. George Pólya observed this to give a generating-function form, which Nicolaas de Bruijn refined and extended, and which allows for weighting the elements. In our example, for any , it gives a polynomial that can be used to compute the number of -gon tile types for any number of colors, without repeating the inspection of fixed elements. The polynomial involves variables raised to the power of the number of cycles of length in a given permutation. Each permutation gives a monomial and those are summed up and divided by the size of the group. In the case of the polynomial is

If we simply substitute each by then we get the number of orbits; for instance, square tiles with 3 colors give 21 orbits. But with colors we can substitute by and then the resulting gives more information. Assigning other *weights* besides to achieves other counting tasks. This article by Nick Baxter serves to introduce a detailed survey by Mollee Huisinga titled “Pólya’s Counting Theory,” which is great for further applications including classifying molecular structures.

We are interested in cases involving succinctly-described permutations of exponential-sized sets . That the general formula for involves the Euler totient function not only of but also divisors of hints at one source of such cases. In our setting, whether fixes an element is decidable in polynomial time, but whether and belong to different orbits—and other predicates about orbits—may not be. Thus the LCFB is instrumental to getting the orbit-counting problem and other functions into .

What applications of the Lemma Cited From Burnside have been noted in complexity theory? We can ask this more generally about applications of the combinatorial double-counting trick, where one way of doing the counting looks hopeless but the other way at least gives a function or difference of functions.

]]>Toutfait.com source |

Marcel Duchamp was a leading French chess player whose career was sandwiched between two forays into the art world. He played for the French national team in five chess Olympiads from 1924 to 1933. He finished tied for fourth place out of fourteen players in the 1932 French championship.

Today we look afresh at some of his *coups* in art and chess and find some unexpected depth.

We say “unexpected” because Duchamp was famous for art that consisted of common objects tweaked or thrown together. He called them readymades in English while writing in French. An example is his 1917 * Fountain*, which Dick and Kathryn saw in Philadelphia last weekend:

SFMOMA replica, Wikimedia Commons source |

Duchamp first submitted this anonymously to a New York exhibition he was helping to organize. When it was refused, he resigned in protest. He then ascribed it to a person named Richard Mutt. *Richard* means “rich person” in French while “R. Mutt” suggests *Armut* which is German for “poverty.” A magazine defended the work by saying that whether Mr. Mutt made the fountain—which came from the J.L. Mott Iron Works—has no importance:

He CHOSE it… [and thus] created a new thought for that object.

The new thought led in 2004 to a poll of 500 art professionals voting *Fountain* “the most influential artwork of the 20th century.” This was ahead of *Guernica*, *Les Demoiselles d’Avignon*, *The Persistence of Memory*, *The Dance*, *Spiral Jetty*, just to name a few works of greater creation effort. It was ahead of everything by Alexander Calder or Andy Warhol or math favorite Maurits Escher for that matter. For Duchamp that was quite a *coup*—which is also French for a move at chess. *Fountain* is also a kind of *coupe*—French for “cup” and also snippet or section.

Michelangelo Buonarroti famously declared that “every block of stone has a figure inside it and the sculptor’s task is to discover it.” Dick and I feel most of our peers would disagree with this about sculpture but agree with the same remark applied to mathematics. As Platonists we believe our theorems had proofs in “the book” from the start and that our chipping away at problems is what discovers them.

The paradox is that we nevertheless experience theoretical research as being as creative as Michelangelo’s artwork or Duchamp’s original painting, *Nude Descending a Staircase*. What accounts for this? We have previously alluded to “builders” versus “solvers” and the imperative of creating good definitions. Builders still need to sense where and when proofs are likely to be available.

Finding a new proof idea is reckoned as the height of creativity despite the idea’s prior existence. This is however rare. Most of us do not invent or re-invent wheels but rather ride wheels we’ve mastered. They may be wheels we learned in school, as pre-fab as Duchamp’s 1913 *Bicycle Wheel*. The creativity may come from learning to deploy the wheels in a new context:

David Gómez (c) “Duchamp a Hamburger Bahnhof” source, license (photo unchanged) |

Neither of the works pictured above is the original version. The originals were lost, as were second versions made by Duchamp. Duchamp’s third *Bicycle Wheel* belongs to MoMA, which is adjacent to Dick and Kathryn’s apartment in Manhattan. The Philadelphia Art Museum has the earliest surviving replica of *Fountain*, certified by Duchamp as dating to 1950. Duchamp blessed fourteen other replicas in the 1960s.

For contrast, Duchamp spent nine years making the original artwork shown behind the board in this crop of an iconic photo:

Cropped from Vanity Fair source |

The nine-foot high construction between glass panes lives in Philadelphia under its English name *The Bride Stripped Bare By Her Bachelors, Even*. The “even” translates the French *même*, which is different from *mème* meaning “meme.” Richard Dawkins coined the latter term in his 1976 book *the Selfish Gene*. Modern Internet “memes” diverge from Dawkins’s meaning but amplify his book’s emphasis on *replication*. Both the isolation of concepts and the replication were anticipated by Duchamp.

A wonderful Matrix Barcelona story has the uncropped photo of Duchamp playing the bared Eve Babitz. It also has a film segment with Duchamp and Man Ray which shows how they viewed the world. Duchamp could paint “retinally” as at left below, but this page explains how his vision of the scene shifted a year later. Then his poster for the 1925 French championship abstracts chess itself:

Composite of sources collected here |

Duchamp’s high-level chess activity stopped before the Second World War broke, but he kept up his interest during it. In 1944-45 he helped organize an exhibition *The Visual Imagery of Chess* at the Julien Levy gallery in midtown Manhattan. One evening featured a *blindfold simultaneous exhibition* by the Belgian-American master George Koltanowski:

Composite of src1, src2 |

This composite photo with leafy allusions shows left-to-right Levy (standing), artist Frederick Kiesler, Duchamp executing a move called out by Koltanowski (facing away), art historian Alfred Barr, Bauhaus artist Xanti Schawinsky, composer Vittorio Rieti, and the married artists Dorothea Tanning and Max Ernst. Plus someone evidently looking for a new game. My “assisted readymade” skirts the edge of *fair use* (non-commercial) *with modification*. The works I combined each have higher creation cost than the objects Duchamp used. Yet there’s no restraint on combining other people’s theorems—of whatever creation cost—with attribution.

Koltanowski kept the seven games entirely in his head, winning six and drawing one, and performed similar feats well into his eighties. Yet Duchamp once beat him in a major tournament—in only 15 moves—when Koltanowski was in his prime and looking at the board.

So how strong was Duchamp? It is hard to tell because Arpad Elo did not create the Elo rating system until the 1950s and because the records of many of Duchamp’s games are lost. Twenty of Duchamp’s tournaments are listed in the omnibus Chessbase compilation but only four have all his games, only four more include as many as five games, and most including the 1923 Belgian Cup lack even the results of his other games. For these eight events, my chess model assesses Duchamp’s “Intrinsic Performance Ratings” (IPRs) as follows:

The error bars are big but the readings are consistent enough to conclude that Duchamp reached the 2000–2100 range but fell short of today’s 2200 Master rank.

My IPRs for historical players have been criticized as too low because today’s players benefit from greater knowledge of the opening, middlegame dynamics, and endgames. My model does not compensate for this—it credits moves that go from brain to board not caring whether preparing with computers at home put them in the brain. However, a comparison with Koltanowski is particularly apt because Elo himself estimated Koltanowski at 2450 based on his play in 1932–1937. My IPR from every available Koltanowski game in those years is 2485 +- 95. When limited to the major—and complete—tournaments that would have most informed Elo’s estimate, it is 2520 +- 100. The latter does much to suggest that Koltanowski might have merited back then the grandmaster title, which he was awarded *honoris causa* in 1988. Koltanowski had 2380 +- 165 in the year 1929, including his loss to Duchamp.

Still, Duchamp’s multiple IPR readings over 2000 earn him the rank of *expert*, which few attain. Duchamp gave himself a different title in 1952:

I am still a victim of chess. It has all the beauty of art—and much more.

Duchamp loved the endgame but is only known to have composed one problem. Fitting for Valentine’s Day, he embellished it with a hand-drawn cupid:

Composite of diagrams from Arena and Toutfait.com source |

Yet unrequited love may be the theme, for there is no solution. Analysis by human masters has long determined the game to be drawn with the confidence of a human mathematical proof. All the critical action can be conveyed in one sequence of moves: 1. Rg7+ Kf2 2. Ke4 h4 3. Kd5 h3 4. Kc6 h2 5. Rh7 Kg2 6. Kc7 Rg8 (or 6…Rf8 or …Re8 or even …Rxb7+ if followed by 7. Kxb7 f5!) 7. b8Q Rxb8 8. Kxb8 h1Q (or 8…f5 first) 9. Rxh1 Kxh1 10. Kc7 f5 11. b6 f4 12. b7 f3 13. b8Q f2 14. Qb1+ Kg2 15. Qe4+ Kg1 16. Qg4+ Kh2 17. Qf3 Kg1 18. Qg3+ Kh1! when 19. Qxf2 is stalemate and no more progress can be made.

The cupid and signature were on one side of the program sheet for an art exhibition titled “Through the Big End of the Opera Glass.” The other side had the board diagram, caption, and mirror-image words saying, “Look through from other side against light.” The upshot arrow was meant as a hint to shoot White’s pawns forward. But by making a mirror image of the position instead, I have found the second of two surprising effects.

The first surprise is that when it comes to verifying the draw with today’s chess programs—which are far stronger than any human player—Duchamp’s position splits them wildly. This is without equipping the programs with endgame tables—just their basic search algorithms.

The Houdini 6 program, which has just begun defending its TCEC championship against a field led by past champions Komodo and Stockfish, takes only 10 seconds on my office machine to reach a “drawn” verdict that it never revises. Here is a *coupe* of its analysis approaching depth 40 *ply*, meaning a nominal basic search 20 moves ahead. That’s enough to see the final stalemate, so Houdini instead tries to box Black in, but by move 4 we can already see Black’s king squirting out. Its latent threat to White’s pawns knocks White’s advantage down to 0.27 of a pawn, which is almost nada:

Komodo stays with the critical line but churns up hours of thinking time while keeping an over-optimistic value for it:

The just-released version 9 of Stockfish, however, gyrates past depth 30, seemingly settles down like Houdini, but then suddenly goes—and stays—bananas:

When Komodo is given only 32MB hash, it gyrates even more wildly until seeming to settle on a +3.25 or +3.26 value at depths 31–34. Then at depth 35 it balloons up to +6.99 and swoons. After 24 hours it is still on depth 35 and has emitted only two checked-down values of +6.12 and +4.00 at about the 8 and 16 hour points.

Now for the second surprise. The mirror-image position at right above changes *absolutely none of the chess logic*. But when we input it to Stockfish 9, with 32MB hash, it gives a serene computation:

What’s going on? The upshot is that the mirrored position’s different squares use a different set of keys in a tabulation hashing scheme. They give a different pattern of *hash collisions* and hence a different computation.

There are two more mirror positions with Black and White interchanged. One is serene (from depth 20 on) but the other blows up at depths 36–40. This is for Stockfish 9 with 32MB hash, “contempt” set to 0, and default settings otherwise. With 512MB hash, *both* blow up. Since both Stockfish 9 and the Arena chess GUI used to take the data are freely downloadable, anyone can reproduce the above and do more experiments.

There is potential high importance because the large-scale behavior of the hash collisions and search may be sensitive to whether the nearly-50,000 bits making up the hash keys are *truly random* or *pseudorandom*. I detailed this and a reproducible “digital butterfly effect” in a post some years ago.

Thus unexpected things happen to computers at high depth in Duchamp’s position. It is not in the Chessbase database, but he may have gotten it “readymade” from playing a game or analyzing one. In all cases we can credit the astuteness of his *choosing* it.

What will be Duchamp’s legacy in the 21st Century? Chess players will keep it growing. Buenos Aires (where he traveled to study chess in 1919), Rio de Janeiro, and Montevideo have organized tournaments in his honor. It was my pleasure to monitor the 2018 Copa Marcel Duchamp which finished last week in Montevideo. This involved getting files of the games from arbiter Sabrina de San Vicente, analyzing them using spare capacity on UB’s Center for Computational Research, and generating ready-made statistical reports for her and the tournament staff to view.

[a few slight fixes and tweaks; added note about contempt=0]