Mathematical_Recreations-Kraitchik-2e

Games 301
in the original game, except that doublets must be placed
crosswise, and no play is allowed from the end of a doublet.
The motives of play are different, however, since a player
scores after each play according to whether the totals at the
free ends of the chain is exactly divisible by 5. If this total
is 0 or is not divisible by 5, the player scores o. If the total
is exactly divisible by 5 the player scores the quotient. (Thus
a total of 20 scores 4.)
The 5-5 is the best domino in this game. Suppose A plays
it. He scores 2. Unless B has the 5-0 (which would give
him 2) he cannot score. Suppose B plays 5-6. If C plays
5-4 he scores 2. If A now plays 4-4 he scores nothing. If B
then plays 6-6 he scores 4. And so on.
The number 3 is sometimes used instead of 5.
There are some interesting properties of the dominoes
themselves.
(n+1)(n+2)
If the largest domino is a n-n, there are 2
dominoes in the set, each number from 0 to n occurs n + 2
times on the faces of the dominoes, and the sum of the numbers
of all the dominoes of the set is therefore (n + 2) . n(n 2+ 1).
Hence also the average value of a domino is n.
Using the rules first laid down, it is always possible to
form a chain containing the whole set of dominoes, provided
n is 1 or an even number, but it is impossible for any other
value of n. When such a chain has been formed for an even
value of n, the numbers at the two ends are the same.
There is a simple recreation based on this property.
Adroitly conceal any domino except a double, say 3-4, and
ask someone to form a proper chain with all the remaining
ones. You can promise him that however he does it, one
end must have a 3 and the other a 4.
One may construct magic squares with dominoes, although
if the order is greater than 3 one must be permitt~d to repeat