## Review: Number Freak: From 1 to 200: The Hidden Language of Numbers Revealed

Every now and then someone offers to send me a book to review. I have a hard time saying no to fun math books. If I find something I like about the book I'll spend the time to write my impressions.

Penguin offered me a copy of Derrick Niederman's new book, Number Freak, to review. Niederman has written a number of books. I hadn't heard of Niederman before so I looked up his Amazon page.

I really like the book. Niederman writes several interesting things about each of the numbers from 1 to 200. What I most like about the book is that many of the number facts lead to fascinating explorations. 16 is my favorite number and here's a great exploration about 16:

Define two sets of numbers A and B as follows:

A={1,4,6,7,10,11,13,16}

B={2,3,5,8,9,12,14,15}It is obvious at a glance that the sets A and B are (1) disjoint and (2) together account for every positive integer from 1 to 16. A second glance reveals that each of the eight pairs {1,2} through {15,16} has precisely one element in A and one in B, with four even numbers and four odd numbers in each set, so that the sum of the members of A equals the sum of the members of B. But what is considerably less obvious is that the sum of the

squaresof the elements of A equals the sum of the squares of the elements of B, and similarly forcubes. This construction is remarkable but turns out to be possible for any power of 2: a 32-number construction uses fourth powers, a 64-number construction incorporates fifth powers, and so on.

Here's another interesting thing to ponder:

It is well-known that 50 = 1^2+7^2 = 5^2+5^2 and is the smallest number that can be expressed as the sum of two squares in two different ways. But there's a geometric interpretation to this fact that isn't quite as familiar.

Can you come up with the geometric interpretation?

These are just two of many fun number facts. If you like recreational Math I don't think you'll be disappointed with this book.

efriqueSeptember 3rd, 2009 - 22:07

“But what is considerably less obvious is that the sum of the squares of the elements of A equals the sum of the elements of B”

Typo: missing “squares of” just near the end there.

SolSeptember 4th, 2009 - 06:11

Good catch. Fixed. Thank you!

notedscholarSeptember 5th, 2009 - 12:25

These sorts of coincidences in abstract space help support the theory that there is intelligent design lurking about.

I’m not saying prove, just support.

NS