Mrs. Sandy McAllister will have to save a tremendous sum out of her
housekeeping allowance if she is to win that sixth present that her
canny husband promised her. And the allowance must be a very liberal one
if it is to admit of such savings. The problem required that we should
find five numbers higher than 36 the units of which may be displayed so
as to form a square, a triangle, two triangles, and three triangles,
using the complete number in every one of the four cases.
Every triangular number is such that if we multiply it by 8 and add 1
the result is an odd square number. For example, multiply 1, 3, 6, 10,
15 respectively by 8 and add 1, and we get 9, 25, 49, 81, 121, which are
the squares of the odd numbers 3, 5, 7, 9, 11. Therefore in every case
where 8x² + 1 = a square number, x² is also a triangular. This point
is dealt with in our puzzle, "The Battle of Hastings." I will now merely
show again how, when the first solution is found, the others may be
discovered without any difficulty. First of all, here are the figures:--
8 × 1² + 1 = 3²
8 × 6² + 1 = 17²
8 × 35² + 1 = 99²
8 × 204² + 1 = 577²
8 × 1189² + 1 = 3363²
8 × 6930² + 1 = 19601²
8 × 40391² + 1 = 114243²