Digital Puzzles 37"Not at all," he replied. "Take another number at random-4913-and the
digits add to 17, the cube of which is 4913."
I did not presume to argue the point with the learned man, but I will just
ask the reader to discover all the other numbers whose cube root is the same
as the sum of their digits. They are so few that they can be counted on
the fingers of one hand.- QUEER DIVISION
The following is a rather curious puzzle. Find the smallest number that,
when divided successively by 45, 454, 4545 and 45454, leaves the remainders
4, 45, 454, and 4,545 respectively. This is perhaps not easy but it affords
a good arithmetical exercise.- THREE DIFFERENT DIGITS
The professor, a few mornings ago, proposed that they should find all those
numbers composed of three different digits such that each is divisible without
remainder by the square of the sum of those digits. Thus, in the case of 112,
the digits sum to 4, the square of which is 16, and 112 can be divided by 16
without remainder, but unfortunately 112 does not contain three different
digits.
Can the reader find all the possible answers?- DIGITS AND CUBES
Professor Rackbrane recently asked his young friends to find all those five-
figure squares such that the number formed by the first two figures added to
that formed by the last two figures should equal a cube. Thus with the square
of 141, which is 19,881, if we add 19 and 81 together we get 100, which
is a square but unfortunately not a cube.
How many solutions are there altogether?- REVERSING THE DIGITS
What number composed of nine figures, if multiplied by 1,2, 3,4, 5, 6, 7,
8, 9, will give a product with 9, 8, 7, 6, 5, 4, 3, 2, 1 (in that order), in the last
nine places to the right?