Games and Mathematics, by David Wells

I recently mentioned The Joy of X as a great gift for the HS student in your life.  Today I would add another, in fact several more, that I think would be an excellent gift for a the young mathematician in your life this Christmas.

I have been a fan of David Wells for several years.  His Penguin Book of Curious and Interesting Mathematics, The Penguin Book of Curious and Interesting Numbers , and The Penguin Dictionary of Curious and Interesting Geometry,have been favorites for a long time.  Now he has just released Games and Mathematics, which may be the best of them all for the talented HS mathematician.




Folks who have followed me for awhile know that I keep up on most of the math books for the masses.  I read and enjoy them for the way they present information, and on rare occasion I find some gem tucked in that I had not known before, as I have in this book. 

In particular, I learned about a theorem by Joeseph Liouville that seems to tie together one of those topics that I always assumed was just an isolated coincidence in advanced math classes.  I apologize to all the kids I let slip by without pointing this one out. I was made even more embarrassed by the fact that the theorem appears in the book Math competition Tutorial (for Grade 6, primary school), which was published by Peking Normal University Publishing Company and written by Sun Ruiqing who is a professor from Beijing Normal University.  

The common topic is the fact that the sum of the first n cubes is the square of the sum of the first n integers.  1³ + 2³ + 3³ + ... + n³ = (1 + 2 + 3 + ... + n)².
What David Wells points out is that this is just a special case of a more general theorem, the one that Liouville discovered.  There are an infinite number of number sets which have the property that the sum of their cubes is equal to the square of their sum.  Liouville discovered that if you take any number and write its divisors, and then the number of factors for each of these divisors, it turns out that that set of numbers has the property mentioned above. 
For example, if we pick the number six, the divisors are 6, 3, 2, and 1.  Now if we make a list of the number of factors that each of these numbers have, we see that :
6 has 4 factors,  1,2,3, and 6
3 has 2 factors, 1 and 3
2 has 2 factors, 1 and 2
and 1 has 1 factor, itself.
And the numbers 4, 2, 2, 1 form a set which has the property that 4³ + 2³ + 2³ + 1³ = 64+ 8+8+1 = 81 = (4+2+2+1) ².

And that ALWAYS works.  Try it on your favorite number.

How does this specialize into the formula memorized by Pre-Calc students everywhere?  Start with a power of a prime, say 3⁵ .   Its factors are 3⁵, 3⁴, 3³, 3², 3¹, and 3⁰ and the number of divisors of each of these numbers is 6, 5, 4, 3, 2, and 1, so by Liouville's mathematical discovery,  1³ + 2³ + 3³ + ... + 6³ = (1 + 2 + 3 + ... + 6)².  Replace the 5 with n-1 and you have the general identity above.

I think a talented young mathematician would love wandering through Well's discussion of this and other ideas that tend to take them way beyond the hum-drum classroom exercises.  And each of the others above is an excellent adventure for them as well.

Make it a Merry Mathematical Christmas for someone, perhaps yourself.