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Amusements in Mathematics

long diagonals. The puzzle is with the remaining cards (without disturbing this
arrangement) to form three more such magic squares, so that each of the four shall add up
to a different sum. There will, of course, be four cards in the reduced pack that will not be
used. These four may be any that you choose. It is not a difficult puzzle, but requires just
a little thought.
The illustration shows eighteen dominoes arranged in the form of a square so that the
pips in every one of the six columns, six rows, and two long diagonals add up 13. This is
the smallest summation possible with any selection of dominoes from an ordinary box of
twenty-eight. The greatest possible summation is 23, and a solution for this number may
be easily obtained by substituting for every number its complement to 6. Thus for every
blank substitute a 6, for every 1 a 5, for every 2 a 4, for 3 a 3, for 4 a 2, for 5 a 1, and for
6 a blank. But the puzzle is to make a selection of eighteen dominoes and arrange them
(in exactly the form shown) so that the summations shall be 18 in all the fourteen
directions mentioned.
Although the adding magic square is of such great antiquity, curiously enough the
multiplying magic does not appear to have been mentioned until the end of the eighteenth
century, when it was referred to slightly by one writer and then forgotten until I revived it
in Tit-Bits in 1897. The dividing magic was apparently first discussed by me in The
Weekly Dispatch in June 1898. The subtracting magic is here introduced for the first time.
It will now be convenient to deal with all four kinds of magic squares together
In these four diagrams we have examples in the third order of adding, subtracting,
multiplying, and dividing squares. In the first the constant, 15, is obtained by the addition
of the rows, columns, and two diagonals. In the second case you get the constant, 5, by
subtracting the first number in a line from the second, and the result from the third. You
can, of course, perform the operation in either direction; but, in order to avoid negative
numbers, it is more convenient simply to deduct the middle number from the sum of the
two extreme numbers. This is, in effect, the same thing. It will be seen that the constant
of the adding square is n times that of the subtracting square derived from it, where n is
the number of cells in the side of square. And the manner of derivation here is simply to
reverse the two diagonals. Both squares are "associated"—a term I have explained in the
introductory article to this department.
The third square is a multiplying magic. The constant, 216, is obtained by multiplying
together the three numbers in any line. It is "associated" by multiplication, instead of by
addition. It is here necessary to remark that in an adding square it is not essential that the
nine numbers should be consecutive. Write down any nine numbers in this way—