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 L5 present she must save the respectable sum of L1,631,432,881.

138.—­THE ARTILLERYMEN’S DILEMMA.

We were required to find the smallest number of cannon balls that we could lay on the ground to form a perfect square, and could pile into a square pyramid.  I will try to make the matter clear to the merest novice.

1 2 3 4 5 6 7 1 3 6 10 15 21 28 1 4 10 20 35 56 84 1 5 14 30 55 91 140

Here in the first row we place in regular order the natural numbers.  Each number in the second row represents the sum of the numbers in the row above, from the beginning to the number just over it.  Thus 1, 2, 3, 4, added together, make 10.  The third row is formed in exactly the same way as the second.  In the fourth row every number is formed by adding together the number just above it and the preceding number.  Thus 4 and 10 make 14, 20 and 35 make 55.  Now, all the numbers in the second row are triangular numbers, which means that these numbers of cannon balls may be laid out on the ground so as to form equilateral triangles.  The numbers in the third row will all form our triangular pyramids, while the numbers in the fourth row will all form square pyramids.