# 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.

406.—THE EIGHTEEN DOMINOES.

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.

SUBTRACTING, MULTIPLYING, AND DIVIDING

MAGICS.

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—

135