This exercise may be
diversified by your specifying, before the sum is named, the particular
place where the figure shall be inserted, to make the number divisible
by 9; for it is exactly the same thing whether the figure be put at
the head of the number, or between any two of its digits.
THE MAGIC HUNDRED.
Two persons agree to take, alternately, numbers less than a given
number, for example, 11 and to add them together till one of them has
reached a certain sum, such as 100. By what means can one of them
infallibly attain to that number before the other? The whole secret
in this consists in immediately making choice of the numbers, 1, 12,
23, 34, and so on, or of a series which continually increases by 11,
up to 100. Let us suppose, that the first person, who knows the game,
makes choice of 1; it is evident that his adversary, as he must count
less than 11, can, at most, reach 11 by adding 10 to it. The first
will then take 1, which will make 12; and whatever number the second
may add, the first will certainly win, provided he continually add the
number which forms the complement of that of his adversary, to 11;
that is to say, if the latter take 8, he must take 3; if 9, he must
take 2; and so on.
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