THE STONEMASON'S PROBLEM.
(
Money Puzzles)
A stonemason once had a large number of cubic blocks of stone in his
yard, all of exactly the same size. He had some very fanciful little
ways, and one of his queer notions was to keep these blocks piled in
cubical heaps, no two heaps containing the same number of blocks. He had
discovered for himself (a fact that is well known to mathematicians)
that if he took all the blocks contained in any number of heaps in
regular order, beginning with the single cube, he could always arrange
those on the ground so as to form a perfect square. This will be clear
to the reader, because one block is a square, 1 + 8 = 9 is a square, 1 +
8 + 27 = 36 is a square, 1 + 8 + 27 + 64 = 100 is a square, and so on.
In fact, the sum of any number of consecutive cubes, beginning always
with 1, is in every case a square number.
One day a gentleman entered the mason's yard and offered him a certain
price if he would supply him with a consecutive number of these cubical
heaps which should contain altogether a number of blocks that could be
laid out to form a square, but the buyer insisted on more than three
heaps and _declined to take the single block_ because it contained a
flaw. What was the smallest possible number of blocks of stone that the
mason had to supply?
Answer:
The puzzle amounts to this. Find the smallest square number that may be
expressed as the sum of more than three consecutive cubes, the cube 1
being barred. As more than three heaps were to be supplied, this
condition shuts out the otherwise smallest answer, 23 cubed + 24 cubed + 25 cubed =
204 squared. But it admits the answer, 25 cubed + 26 cubed + 27 cubed + 28 cubed + 29 cubed = 315 squared. The
correct answer, however, requires more heaps, but a smaller aggregate
number of blocks. Here it is: 14 cubed + 15 cubed + ... up to 25 cubed inclusive, or
twelve heaps in all, which, added together, make 97,344 blocks of stone
that may be laid out to form a square 312 x 312. I will just remark that
one key to the solution lies in what are called triangular numbers. (See
pp. 13, 25, and 166.)
Informational
