THE TROUBLESOME EIGHT.
(
Magic Squares Problem.)
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 illustration 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 one of the corners. The puzzle is to
construct the magic square, under the same conditions, with the 8 in the
position shown.
Answer:
[Illustration:
+---+---+---+
| 41/2| 8 | 21/2|
+---+---+---+
| 3 | 5 | 7 |
+---+---+---+
| 71/2| 2 | 51/2|
+---+---+---+
]
The conditions were to place a different number in each of the nine
cells so that the three rows, three columns, and two diagonals should
each add up 15. Probably the reader at first set himself an impossible
task through reading into these conditions something which is not
there--a common error in puzzle-solving. If I had said "a different
figure," instead of "a different number," it would have been quite
impossible with the 8 placed anywhere but in a corner. And it would have
been equally impossible if I had said "a different whole number." But a
number may, of course, be fractional, and therein lies the secret of the
puzzle. The arrangement shown in the figure will be found to comply
exactly with the conditions: all the numbers are different, and the
square adds up 15 in all the required eight ways.
