(1 × 3) + (3 × 1) = 6 (8 × 1) + (3 × 3) = 17
(1 × 17) + (3 × 6) = 35 (8 × 6) + (3 × 17) = 99
(1 × 99) + (3 × 35) = 204 (8 × 35) + (3 × 99) = 577
and so on. Look for the numbers in the table above, and the method will
explain itself.
Thus we find that the numbers 36, 1225, 41616, 1413721, 48024900, and
1631432881 will form squares with sides of 6, 35, 204, 1189, 6930, and
40391; and they will also form single triangles with sides of 8, 49,
288, 1681, 9800, and 57121. These numbers may be obtained from the last
column in the first table above in this way: simply divide the numbers
by 2 and reject the remainder. Thus the integral halves of 17, 99, and
577 are 8, 49, and 288.
All the numbers we have found will form either two or three triangles at
will. The following little diagram will show you graphically at a glance
that every square number must necessarily be the sum of two triangulars,
and that the side of one triangle will be the same as the side of the
corresponding square, while the other will be just 1 less.
[Illustration
+-----------+
+---------+ |. . . . ./.|
|. . . ./.| |. . . ./. .|
|. . ./. .| |. . ./. . .|
|. ./. . .| |. ./. . . .|
|./. . . .| |./. . . . .|
/. . . . .| /. . . . . .|
+---------+ +-----------+
]
Thus a square may always be divided easily into two triangles, and the
sum of two consecutive triangulars will always make a square. In numbers
it is equally clear, for if we examine the first triangulars--1, 3, 6,
10, 15, 21, 28--we find that by adding all the consecutive pairs in turn
we get the series of square numbers--9, 16, 25, 36, 49, etc.
The method of forming three triangles from our numbers is equally
direct, and not at all a matter of trial. But I must content myself with
giving actual figures, and just stating that every triangular higher
than 6 will form three triangulars. I give the sides of the triangles,
and readers will know from my remarks when stating the puzzle how to
find from these sides the number of counters or coins in each, and so
check the results if they so wish.
+----------------------+-----------+---------------+-----------------------+
| Number | Side of | Side of | Sides of Two | Sides of Three |
| | Square. | Triangle. | Triangles. | Triangles. |
+------------+---------+-----------+---------------+-----------------------+
| 36 | 6 | 8 | 6 + 5 | 5 + 5 + 3 |
| 1225 | 35 | 49 | 36 + 34 | 33 + 32 + 16 |
| 41616 | 204 | 288 | 204 + 203 | 192 + 192 + 95 |
| 1413721 | 1189 | 1681 | 1189 + 1188 | 1121 + 1120 + 560 |
| 48024900 | 6930 | 9800 | 6930 + 6929 | 6533 + 6533 + 3267 |
| 1631432881 | 40391 | 57121 | 40391 + 40390 | 38081 + 38080 + 19040 |
+------------+---------+-----------+---------------+-----------------------+
I should perhaps explain that the arrangements given in the last two
columns are not the only ways of forming two and three triangles. There
are others, but one set of figures will fully serve our purpose. We thus
see that before Mrs. McAllister can claim her sixth £5 present she must
save the respectable sum of £1,631,432,881.