Answers to Problems
Page 120:
Take a pair (m/n) in the sequence for √2, where n is even, so m^{2} − 2 n^{2} = 1. Then ((m − 1)/2 + n, (m + 1)/2 + n, m + n) is a Pythagorean triple with two sides differing by 1.
 The picture immediately gives the explanation of how to add the numbers from 1 to 9. Two such triangles make a rectangle of 9 × 10 squares, and half of this is 45. To add the numbers from 1 to 100, take half of 100 × 101, which is 5050.

Page 121:
If the triangular number s(s+1)/2 is square, then either s+1 is square and s/2 is square, or s is square and (s+1)/2 is square.
These are equivalent to taking s = m^{2} − 1 when
m^{2} − 1 = 2n^{2}, and
s= m^{2} when m^{2}+ 1 = 2n^{2}.
We know the solutions to these equations from the approximations to √2.
Thus the sequence of possible numbers s runs:
1^{2}, 3^{2} − 1, 7^{2}, 17^{2} − 1, 41^{2}, 99^{2} − 1...
where the numbers 1, 3, 7, 17, 41, 99... come from the approximations to √2.
These give the triangular numbers 1, 36, 1225, 41616, 1413721, 48024900...
