This is an excellent book with two-color illustrations. It starts out with a discussion of numbers systems, and with instances of non-decimal systems that we encounter in our daily lives. It goes on to discuss some basic number theory. It gives an algorithm for generating perfect numbers:

* let S be the sum of the powers of 2 from e = 0 to e = n
* if S is a prime, then S * 2^n = P is a perfect number.
The factors of P are the powers of 2 up to the n power and S * each of these powers of 2 (except the last).
Note that the sum of
2^0 + ... + 2^n = 2^(n + 1) - 1 = S
and S * each of these powers of 2 = S * (2^n - 1) = S * 2^n - S.

It discusses amicable numbers and magic squares, which of course have something to do with triangular numbers. The sum of each row or column in an n * n square is obviously T(n^2) / n =
(n^2 (n^2 + 1) / 2) / n = n (n^2 + 1) / 2. It discusses a grating method for multiplication which takes up more space than the standard method taught in grade school, but which is also a bit more organized, as well as the method of multiplying by constructing powers of 2.

Two number tricks are presented. One is an amusing implementation of binary search in which each card divided the set of all numbers between 0 and 31 into those which have a 1 bit in the nth place and those that do not. There is a card trick which involves a permutation, probably best expressed by:

(1, 1) -> (1, 1)
(1, 2) -> (1, 3)
(1, 3) -> (2, 3)
(2, 1) -> (1, 2)
(2, 2) -> (2, 2)
(2, 3) -> (3, 2)
(3, 1) -> (2, 1)
(3, 2) -> (3, 1)
(3, 3) -> (3, 3)

The pairs represent the position of cards in a three-by-three grid. The permutation is the result of picking all the cards up on the diagonal, starting from the upper left and working upward, and then laying them out by rows. So,

on the pickup, the order of cards by the coordinates is:
(1, 1)
(2, 1)
(1, 2)
(3, 1)
(2, 2)
(1, 3)
(3, 2)
(2, 3)
(3, 3)

on layout, the new order is

(1, 1) (2, 1) (1, 2)
(3, 1) (2, 2) (1, 3)
(3, 2) (2, 3) (3, 3)

The three cards on the diagonal from upper-left have remained the same and will continue to do so. The other cards rotate in a clockwise direction one step with each move. ( )