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Amusements in Mathematics

Problems Concerning Games
"The little pleasure of the game."
Every game lends itself to the propounding of a variety of puzzles. They can be made, as
we have seen, out of the chessboard and the peculiar moves of the chess pieces. I will
now give just a few examples of puzzles with playing cards and dominoes, and also go
out of doors and consider one or two little posers in the cricket field, at the football
match, and the horse race and motor-car race.
It will be seen that I have played six dominoes, in the illustration, in accordance with the
ordinary rules of the game, 4 against 4, 1 against 1, and so on, and yet the sum of the
spots on the successive dominoes, 4, 5, 6, 7, 8, 9, are in arithmetical progression; that is,
the numbers taken in order have a common difference of 1. In how many different ways
may we play six dominoes, from an ordinary box of twenty-eight, so that the numbers on
them may lie in arithmetical progression? We must always play from left to right, and
numbers in decreasing arithmetical progression (such as 9, 8, 7, 6, 5, 4) are not
Here is a new little puzzle that is not difficult, but will probably be found entertaining by
my readers. It will be seen that the five dominoes are so arranged in proper sequence (that
is, with 1 against 1, 2 against 2, and so on), that the total number of pips on the two end
dominoes is five, and the sum of the pips on the three dominoes in the middle is also five.
There are just three other arrangements giving five for the additions. They are: —
(1—0) (0—0) (0—2) (2—1) (1—3)
(4—0) (0—0) (0—2) (2—1) (1—0)
(2—0) (0—0) (0—1) (1—3) (3—0)
Now, how many similar arrangements are there of five dominoes that shall give six
instead of five in the two additions?
It will be seen in the illustration that the full set of twenty-eight dominoes is arranged in
the form of a square frame, with 6 against 6, 2 against 2, blank against blank, and so on,
as in the game. It will be found that the pips in the top row and left-hand column both add
up 44. The pips in the other two sides sum to 59 and 32 respectively. The puzzle is to
rearrange the dominoes in the same form so that all of the four sides shall sum to 44.