e 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?

136.--THE SULTAN'S ARMY.

A certain Sultan wished to send into battle an army that could be formed
into two perfect squares in twelve different ways. What is the smallest
number of men of which that army could be composed? To make it clear to
the novice, I will explain that if there were 130 men, they could be
formed into two squares in only two different ways--81 and 49, or 121
and 9. Of course, all the men must be used on every occasion.

137.--A STUDY IN THRIFT.

Certain numbers are called triangular, because if they are taken to
represent counters or coins they may be laid out on the table so as to
form triangles. The number 1 is always regarded as triangular, just as 1
is a square and a cube number. Place one counter on the table--that is,
the first triangular number. Now place two more counters beneath it, and
you have a triangle of three counters; therefore 3 is triangular. Next
place a row of three more counters, and you have a triangle of six
counters; therefore 6 is triangular. We see that every row of counters
that we add, containing just one more counter than the row above it,
makes a larger triangle.

Now, half the sum of any number and its square is always a triangular
number. Thus half of 2 + 2 squared = 3; half of 3 + 3 squared = 6; half of 4 +
4 squared = 10; half of 5