"They counted the cats in Winchester,
Which number a third of a square.
If a quarter were gone,
Just a cube would stay on.
How many, at least, must be there?"
Some clarification:
The number of cats in Winchester is a perfect square divided by three.
If one-fourth of the cats left town, then the number of cats would be a perfect cube.
What is the least number of cats in Winchester?
Solution to the Problem:
The smallest number of cats is 972.
Let c be the number of cats.
Then c = (1/3)n^2 for some integer n, which makes n divisible by 3.
Let n = 3m for some integer m.
Then, we get c = 3m^2.
When a quarter is gone we are left with (3/4)c = (9/4)m^2.
This number must be a cube, say, k^3 for some integer k.
We then get m^2 = (4/9)k^3.
Or, m = (2/3)k^(3/2), which makes k divisible by 3.
The smallest such k is 9, which makes m = 18 and c = 972.
Thus, the smallest number of cats is 972.
The cat in the top picture is Derby -- my daughter's cat who
drinks from the spigot. Our cat is the grey one posing with the
orange wooden cat.