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.

[Illustration]

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.

[Illustration:  ADDING SUBTRACTING MULTIPLYING DIVIDING]

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—­

1    3    5
4    6    8
7    9   11

so that the horizontal differences are all alike and the vertical differences also alike (here 2 and 3), and these numbers will form an adding magic square.  By making the differences 1 and 3 we, of course, get consecutive numbers—­a particular case, and nothing more.  Now, in the case of the multiplying square we must take these numbers in geometrical instead of arithmetical progression, thus—­