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A Curious Inversion

September 28, 2015

The math of “The Curious Incident of the Dog in the Night-Time”

Mark Haddon wrote the book, The Curious Incident of the Dog in the Night-Time, which was Unknownpublished in 2003. It is about an autistic 15 year-old boy, who is a math savant, and who solves a mystery, in spite of his limitations in relating to people.

Today I want to comment on a minor historical inversion at the end of both the book and the current play that is based on Haddon’s book.

I had the great pleasure to see the play recently and found it an amazing experience. The story is told solely from the point of view of an autistic boy, named Christopher Boone. Amazon says:

Christopher John Francis Boone knows all the countries of the world and their capitals and every prime number up to 7,057. He relates well to animals but has no understanding of human emotions. He cannot stand to be touched. And he detests the color yellow.

I re-read the book days before seeing the play, and was unable to even imagine how the play could capture the feel of the book. But they did it. A New York Times review says:

Such a state of being is conjured with dazzling effectiveness in “The Curious Incident of the Dog in the Night-Time,” which opened on Sunday night at the Ethel Barrymore Theater. Adapted by Simon Stephens from Mark Haddon’s best-selling 2003 novel about an autistic boy’s coming-of-age, this is one of the most fully immersive works ever to wallop Broadway.

It was definitely a wallop. Both the book and the play end with a nice geometric problem. In both the answer, which is a proof, is left out of the main part. It is detailed in the book in an appendix; in the play it is delivered by Christopher after all the curtain calls. An “appendix” to a play—what a clever idea.

The Problem

So let’s start by stating the geometric problem from both the book and play.

Prove the following: A triangle with sides that can be written in the form {n^{2}+1}, {n^{2}-1}, and {2n}, where {n>1}, is a right triangle.

The Proof

The proof starts by showing that {n^{2}+1} is the longest side; this uses {n>1}. Then it proves that

\displaystyle  (2n)^{2} + (n^{2}-1)^{2} = (n^{2} + 1)^{2}.

It then states that by the Pythagorean Theorem the triangle is a right one.

But this is inverted.

Some History

The famous Pythagorean theorem states:

Theorem: Let the sides of a right triangle be {a,b,c} with {c} the largest. Then {a^{2} + b^{2} = c^{2}}.

The proof of the problem from the book uses not this theorem—this is the inversion. Rather it uses the converse: For any triangle with sides {a, b, c}, if {a^{2} + b^{2} = c^{2}}, then it is a right triangle.

Happily this converse of the Pythagorean theorem is also a theorem. Indeed Euclid already had proved it. I must admit that I was not sure it was a theorem.

At the play I heard the problem for the first time, since when I read the book I skipped the appendix. As Christopher proved the theorem for the audience, I was almost ready to raise my hand—as if it were a seminar talk—and ask: isn’t there a potential issue with the proof, since it relies on the converse not the actual Pythagorean Theorem? Then I realized this wasn’t a lecture hall, and left the theater quietly.

A Curiosity?

The proof of the converse is not hard, but it is definitely a different theorem. What’s curious, however, is that its proof uses the original Pythagrean theorem. Here is Euclid’s proof as relayed by Wikipedia from this source:

Let {ABC} be a triangle with side lengths {a}, {b}, and {c}, with {a^{2} + b^{2} = c^{2}}. Construct a second triangle with sides of length {a} and {b} containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length {c = \sqrt{a^{2} + b^{2}}}, the same as the hypotenuse of the first triangle. Since both triangles’ sides are the same lengths {a}, {b} and {c}, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths {a} and {b} in the original triangle is a right angle.

So here we have a proof of the ({\Longleftarrow}) direction of an equivalence whose proof uses the ({\Longrightarrow}) direction. How common is that?

Open Problems

Did you know that the Pythagorean Theorem was an “if and only if theorem?” I did not. Are there other notable cases of equivalences where the proof from the “Book” of the converse direction uses the forward direction?

Frogs and Lily Pads and Discrepancy

September 24, 2015

A breakthrough result shows the power of “almost”

Cropped from Quanta Magazine source

Terry Tao has done it again. In two beautiful papers with modest titles, he has evidently proved the famous Discrepancy Conjecture (DC) of Paul Erdős. This emerged from discussion of his two earlier posts this month on his blog. They and his 9/18 announcement post re-create much of the content of the papers.

Today we wish to present just the statement of his new result in a vivid manner and some meta-observations on how he arrived at it. Read more…

Taming Some Inequalities

September 19, 2015

As used to solve a classic problem about distinguishing distributions

Composite of src1, src2

Gregory Valiant and Paul Valiant are top researchers who are not unrelated to each other. Families like the Valiants and Blums could be a subject for another post—or how to distinguish from those who are unrelated.

Today Ken and I wish to talk about a wonderful paper of theirs, “An Automatic Inequality Prover and Instance Optimal Identity Testing.”
Read more…

A Polynomial Growth Puzzle

September 12, 2015

Correcting an erratum in our quantum algorithms textbook

Cropped from source

Paul Bachmann was the first person to use {O}-notation. This was on page 401 of volume 2 of his mammoth four-part text Analytic Number Theory, which was published in Germany in 1894. We are unsure, however, whether he defined it correctly.

Today we admit that we got something wrong about {O}-notation in an exercise in our recent textbook, and we ask: what is the best way to fix it?
Read more…

Open Problems That Might Be Easy

September 3, 2015

A speculation on the length of proofs of open problems

Broad Institute source

Nick Patterson is one of the smartest people I have ever known.

Today I would like to talk about something he once said to me and how it relates to solving open problems.
Read more…

How Joe Traub Beat the Street

August 31, 2015

An insight into the computation of financial information

Columbia memorial source

Joseph Traub passed away just a week ago, on August 24th. He is best known for his computer science leadership positions at CMU, Columbia, CSTB, the Journal of Complexity—they all start with “C.” CSTB is the Computer Science and Telecommunications Board of the National Academies of Science, Engineering, and Medicine. At each of these he was the head and for all except Carnegie-Mellon he was the first head—the founder.

Today Ken and I wish to highlight one technical result by Traub and his co-workers that you may not know about.
Read more…

Cancellation is a Pain

August 27, 2015

How to avoid the pain of estimating tough sums

Cricketing source

Andrew Granville is a number theorist, who has written—besides his own terrific research—some beautiful expository papers, especially on analytic number theory.

Today Ken and I wish to talk about his survey paper earlier this year on the size of gaps between consecutive primes.
Read more…


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