Mathematical_Recreations-Kraitchik-2e

60 Mathematical Recreations
If there are n persons, subtract the number 5· . ·5 (composed
of n digits 5) from the number obtained by the last person,
and divide by 10. The result will be a number whose digits
are the chosen numbers in order.
5. A person is presented with a large list of numbers from
which he is to select two without telling what they are. He
is then instructed to add the chosen numbers, cancel any
digit in the result, add the uncanceled digits and give the
total. What digit was canceled?
Solution: The list is made up only of numbers exactly divisible
by 9, so the sum of its digits' is a multiple of 9. To
find the digit that was canceled one has only to subtract the
stated total from the next larger multiple of 9. To avoid the
possible ambiguity arising when the digit canceled is 0 or 9,
the numbers put in the list should be so chosen that the digit
o does not appear in the sum of any two of them.
6. For variety one may arrange to have the multiple of 9
produced by someone else. Let him think of any number he
pleases. Tell him to form the sum of its digits, subtract this
sum from the given number, multiply the result by any number
he wishes, and then cancel any digit except o. From the
sum of the remaining digits the canceled digit may be found
as before.
Here is another variant. Tell the person to add to the
number selected 8 times the sum of its digits, multiply the
result by an arbitrary number, cancel one digit (other than 0)
in the product and tell the sum of the remaining digits. The
canceled digit can be found as before. Instead of 8, one less
than any multiple of 9 may be used.
Again, the first operation may be varied by asking the person
to subtract from his number any number formed by rearranging
its digits. Or he may add to his number two such
rearrangements (which yields a multiple of 3) and then square