62 Arithmetic & Algebraic Problems- PROPORTIONAL REPRESENTATION
When stopping at Mangleton-on-the-Bliss the Crackhams found the inhab-
itants of the town excited over some little local election. There were ten
names of candidates on a proportional representation ballot. Voters should
place No. I against the candidate of their first choice. They might also place
No.2 against the candidate of their second choice, and so on until all the ten
candidates have numbers placed against their names.
The voters must mark their first choice, and any others may be marked or
not as they wish. George proposed that they should discover in how many
different ways the ballot might be marked by the voter.- A QUESTION OF CUBES
Professor Rackbrane pointed out one morning that the cubes of successive
numbers, starting from I, would sum to a square number. Thus the cubes of
1,2,3 (that is, 1,8,27), add to 36, which is the square of 6. He stated that if
you are forbidden to use the I, the lowest answer is the cubes of 23, 24, and
25, which together equal 2042 • He proposed to seek the next lowest number,
using more than three consecutive cubes and as many more as you like, but
excluding I.- TWO CUBES
"Can you find," Professor Rackbrane asked, "two consecutive cube num-
bers in integers whose difference shall be a square number? Thus the cube of
3 is 27, and the cube of 2 is 8, but the difference, 19, is not here a square
number. What is the smallest possible case?"
- CUBE DIFFERENCES
If we wanted to find a way of making the number 1,234,567 the difference
between two squares, we could at once write down 617,284 and 617,283-
a half of the number plus'll and minus Ih respectively to be squared. But it
will be found a little more difficult to discover two cubes the difference
of which is 1,234,567.