Archive for the ‘Juxtapositions’ Category

Juxtaposition: Millionaire Triangles

February 24, 2009

There was a math question on Who Wants to Be a Millionaire yesterday!  According to this transcript of the February 23 game, this was the $50,000 question:

Named for the mathematician who designed it, a famous “pyramid” of numbers that starts with the number one on top is called what?
A. Fibonacci’s triangle
B. Pythagoras’s triangle
C. Pascal’s triangle
D. Fermat’s triangle

Fibonacci's TriangleThe correct answer is Pascal’s triangle.  But it got me thinking, what would the other triangles look like?  We’ve actually posted before on Fibonacci’s Triangle:  as envisioned by Doug Ensley, it’s a triangle with Fibonacci numbers down each side and with each interior number equal to the sum of the two numbers above it (as with Pascal’s triangle).  This is also called Hosoya’s triangle, according to wikipedia (named after Haruo Hosoya, who wrote about it in The Fibonacci Quarterly in 1976).

fermat-triangleSo what about Fermat’s triangle — what would that look like?  Since Fermat numbers are numbers of the form 2^{2^n} + 1, a natural adaptation to Fibonacci’s triangle would be to have the Fermat numbers on the outside, and to use sums of numbers to fill in the inside.  This looks like of funny, though — the numbers on the outside grow much faster than those on the inside.

What has me stumped is the notion of Pythagoras’s Triangle.  Would it have something to do with Pythagorean triples?  Maybe it would be a tetrahedron rather than a flat triangle, with a 1 on top, and then a Pythagorean triple beneath it — say the iconic 3, 4, 5 — and then…then I’m stumped.  What would come next?  Is there even an ordering for Pythagorean triples?  (Well, maybe:  TwoPi wrote earlier about how every Pythagorean triple can be written as (2uv, u^2-v^2, u^2+v^2), for positive integers u and v, so we could put an ordering on u and v and retrict ourselves to the primitive triples.)  But I’m not sure how to generate the entire tetrahedron.

Then again, for a Pythagorean triangle, perhaps we could just draw a right triangle and call it a day.

Incidentally, the link to the transcript also has commentary about whether Pascal’s triangle was the only reasonable answer.  The consensus seems to be YES, if only because the question referred to a “famous” triangle “pyramid”.

The Math of Chess

September 16, 2008
"Excelsior" by Sam Loyd

White to move, mate in 5. (Sam Loyd, 1861)

It probably comes as no surprise to anyone that knows me that I enjoy playing chess. Indeed, chess is a common hobby among mathematicians (as are Go, juggling, and magic). But I was still amazed to see that a book I was reading, Chess Endings for the Practical Player by Ludek Pachman, was translated by someone named Hardy, and edited by someone named John E. Littlewood. I thought, “No way.” Hardy and Littlewood do chess? The same two that gave us this?

Well, I was right. The Hardy is Otto Hardy, not G.H. Hardy, and the Littlewood is John Eric Littlewood, not John Edensor Littlewood. But wow, close huh?

Had I been wrong — had the Hardy and Littlewood actually worked together on the book — it still shouldn’t have been shocking. It is fairly common to find mathematicians that study and/or play chess. Former chess world champion Max Euwe had his Ph.D. in mathematics, as do GMs John Nunn and Jon Speelman. (Nunn is also a GM chess problem solver – one of only three in history.) Noam Elkies (the youngest full tenured professor in the history of Harvard – at 26!) has published several papers on the mathematics of chess, and earned his solving GM title in 2001.

Math and chess have long been associated with one another. Some famous problems include the Knight’s Tour (any early version of which appeared in the Sanskrit poem Kâvyâlankâra by Rudrata ca. 900):

An open knight's tour.

the Eight Queens Problem, and the Mutilated Chessboard Problem (posed by Martin Gardner). There’s even a book called Mathematics and Chess.

Not all mathematical study of chess is purely recreational. The so-called rook polynomials have found applications in matrix theory and number theory. There’s even a Rook Reciprocity Theorem!

Check out Wikipedia’s Chess and Mathematics category for even more connections between the two, and also see Mathworld’s pages related to chess and math. Finally, no discussion of chess would be complete without a reference to xkcd’s comic that inspired these amazing photos.

Juxtapositions: Song Charts

February 26, 2008

How’s this for a concept: take a popular song, and create a graph or chart that communicates the content of the song lyric or title. There’s a slew of these over on Flikr. My two favorites are:

#1) [featuring a lovely Venn diagram] (by “moved”)


and #2) [not at all mathematical, but funny nonetheless] (by “brianmn”)


Well, ok, maybe these are my favorites because they’re nearly the only ones where I knew what song they referred to (out of the nearly 200 examples on Flickr). I guess I’m just too old to be down with the hep music the cool kids listen to these days. **

Challenge: Create your own, and post them in the comments! (Bonus points for songs or charts with mathematical references.)

This graph-making exercise is vaguely reminiscent of my favorite powerpoint example floating the interweb: Peter Norvig’s devastating ppt adaptation of Abe Lincoln’s Gettysburg Address.

** True: I received my AARP membership card in the mail today. No foolin’! Heck of a thing for them to send out en masse to folk on some mailing list or other in hopes of drumming up support and new members. Makes me wonder what mailing list I’m on that AARP purchased…

Charts used with permission of boyshapedbox, who had posted them to flickr.

Juxtapositions: Sudoku Lotto

January 21, 2008

There I was, a mathematician in the grocery checkout line. Inexplicably, I found my eye drawn to the scratch-off lottery ticket machine. Unlike the proverbial punk in a Dirty Harry movie, I did feel lucky, and I surreptitiously slipped a couple bucks out of my wallet, hoping the cashier wouldn’t notice and take me for a fool.

One of the current scratch-off games offered by the New York Lottery is a SuDoKu game.


Juxtapositions: Killer Sudoku

December 23, 2007

killer-sudoku.jpgIn a comment on an earlier post aboout Sudoku and Kakuro, Batman mentioned a combination of the two games known alternately as Killer Sudoku, Samunamupure or Sum Number Place. As in traditional Sudoku, Killer Sudoku is played on a 9×9 grid in which the digits 1-9 are placed so that each digit appears once in each row, each column, and each 3×3 grid (nonet). As in Kakuro, groups of cells (cages) add to given sums, and within each cage the digits must be distinct. Click to read more and see an enlargement of the game pictured here at the left.

Juxtapositions: The Rubik’s Hypercube

November 18, 2007

Rubik's cube variationsMany people are familiar with the Rubik’s Cube, the 3x3x3 cube with colored faces that can be moved out of position and then, ideally, twisted back into place. This toy, originally called the Magic Cube, was invented by Ernő Rubik in 1974 while he was a lecturer at the Academy of Applied Arts and Crafts in Budapest, Hungary. In 1980 the cube made its way to other countries and spawned, among other things, a one-season cartoon series Rubik, the Amazing Cube in which some kids use a magical come-to-life Rubik’s Cube to solve mysteries, while avoiding the mandatory evil magician who wants to steal the Cube.

Click to read more and watch the Cartoon Intro, because you know you want to

Juxtapositions: Fibonacci’s Triangle

November 17, 2007

What happens when you take two different ideas and put them together? That’s a driving question in mathematical research, and it can lead to some interesting and entertaining results.

Pascal’s Triangle One such juxtaposition is the concept of Fibonacci’s Triangle: a combination of the Fibonacci Sequence (1, 1, 2, 3, 5, 8, … where each number is the sum of the previous two) and Pascal’s Triangle*, shown to the left, which is created by placing “1”s along the outside and then filling the inside by adding two adjacent numbers and placing the sum between them on the next row.

Fibonacci’s Triangle

In Fibonacci’s Triangle, according to Doug Ensley (see below), the string of “1”s along the outside is replaced by the Fibonacci sequence. The inside is filled in the same way as in Pascal’s triangle, with adjacent numbers in a row added together and the sum placed between them in the next row.

Sierpinski’s TriangleThere are many questions that can be asked about Fibonacci’s triangle: Is there an explicit formula for the entries in each row the way there is for Pascal’s triangle? Does a Sierpinski-like pattern develop when you shade in the odd numbers the way it does for Pascal’s triangle? The answer to the first question is yes: see Doug Ensley’s article, “Fibonacci’s Triangle and Other Abominations” in the September 2003 issue of Math Horizons. For the second question, the best solution is to draw it out and see for yourself!

*You can see the picture in a 1665 copy of Pascal’s book Traité du triangle arithmétique here; other images are available from the Cambridge University Library Digital Image Collection.