THE TEN COUNTERS.
(
Money Puzzles)
In this case we use the nought in addition to the 1, 2, 3, 4, 5, 6, 7,
8, 9. The puzzle is, as in the last case, so to arrange the ten counters
that the products of the two multiplications shall be the same, and you
may here have one or more figures in the multiplier, as you choose. The
above is a very easy feat; but it is also required to find the two
arrangements giving pairs of the highest and lowest products possible.
Of course every counter must be used, and the cipher may not be placed
to the left of a row of figures where it would have no effect. Vulgar
fractions or decimals are not allowed.
Answer:
As I pointed out, it is quite easy so to arrange the counters that they
shall form a pair of simple multiplication sums, each of which will give
the same product--in fact, this can be done by anybody in five minutes
with a little patience. But it is quite another matter to find that pair
which gives the largest product and that which gives the smallest
product.
Now, in order to get the smallest product, it is necessary to select as
multipliers the two smallest possible numbers. If, therefore, we place 1
and 2 as multipliers, all we have to do is to arrange the remaining
eight counters in such a way that they shall form two numbers, one of
which is just double the other; and in doing this we must, of course,
try to make the smaller number as low as possible. Of course the lowest
number we could get would be 3,045; but this will not work, neither will
3,405, 3,45O, etc., and it may be ascertained that 3,485 is the lowest
possible. One of the required answers is 3,485 x 2 = 6,970, and 6,970 x
1 = 6,970.
The other part of the puzzle (finding the pair with the highest product)
is, however, the real knotty point, for it is not at all easy to
discover whether we should let the multiplier consist of one or of two
figures, though it is clear that we must keep, so far as we can, the
largest figures to the left in both multiplier and multiplicand. It will
be seen that by the following arrangement so high a number as 58,560 may
be obtained. Thus, 915 x 64 = 58,560, and 732 x 80 = 58,560.
Informational
