also Associated, because in it every number, if added to the number that is equidistant, in a straight line, from the centre gives 17. Thus, 1+16, 2+15, 3+14, etc. The fourth example, considered the most "perfect" of all, is a Nasik. Here all the broken diagonals sum to 34. Thus, for example, 15 +14 + 2 + 3, and 10 + 4 + 7 + 13, and 15 + 5 + 2 + 12. As a consequence, its properties are such that if you repeat the square in all directions you may mark off a square, 4×4, wherever you please, and it will be magic.
The following table not only gives a complete enumeration under the four forms described, but also a classification under the twelve graphic types indicated in the diagrams. The dots at the end of each line represent the relative positions of those complementary pairs, 1 + 16, 2 + 15, etc., which sum to 17. For example, it will be seen that the first and second magic squares given are of Type VI., that the third square is of Type HI., and that the fourth is of Type I. Èdouard Lucas indicated these types, but he dropped exactly half of them and did not attempt the classification.
| Nasik (Type I.) | 48 | |||
| Semi-Nasik | (Type II., Transpositions of Nasik) | 48 | ||
| Semi Nasik„ | (Type III., Associated) | 48 | ||
| Semi Nasik„ | (Type IV.) | 96 | ||
| Semi Nasik„ | (Type V.) | 96 | 192 | |
| Simple. | (Type VI.) | 96 | 384 | |
| Simple„ | (Type VI.) | 208 | ||
| Simple„ | (Type VII.) | 56 | ||
| Simple„ | (Type VIII.) | 56 | ||
| Simple„ | (Type IX.) | 56 | ||
| Simple„ | (Type X.) | 56 | 224 | |
| Simple„ | (Type XI.) | 8 | ||
| Simple„ | (Type XII.) | 8 | 16 | 448 |
| 880 |
It is hardly necessary to say that every one of these squares will produce seven others by mere reversals and reflections, which we do not count as different. So that there are 7,040 squares of this order, 880 of which are fundamentally different.
An infinite variety of puzzles may be made introducing new conditions into the magic square. In The Canterbury Puzzles I have given examples of such squares with coins, with postage stamps, with cutting-out conditions, and other tricks. I will now give a few variants involving further novel conditions.
399.—THE TROUBLESOME EIGHT.
Nearly everybody knows that a "magic square" is an arrangement of numbers in the form of a square so that every row, every column, and each of the two long diagonals adds up alike. For example, you would find little difficulty in merely placing a different number in each of the nine cells in the illustra- tion so that the rows, columns, and diagonals shall all add up 15. And at your first attempt you will probably find that you have an 8 in
| 8 | ||
one of the corners. The puzzle is to construct the magic square, under the same conditions, with the 8 in the position shown.
400.— THE MAGIC STRIPS.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
I happened to have lying on my table a number of strips of cardboard, with numbers printed on them from 1 upwards in numerical order. The idea suddenly came to me, as ideas have a way of unexpectedly coming, to make a little puzzle of this. I wonder whether many readers will arrive at the same solution that I did.
Take seven strips of cardboard and lay them together as above. Then write on each of them the numbers 1, 2, 3, 4, 5, 6, 7, as shown, so that the numbers shall form seven rows and seven columns.
Now, the puzzle is to cut these strips into the fewest possible pieces so that they may be placed together and form a magic square, the seven rows, seven columns, and two diagonals adding up the same number. No figures may
