"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 upper left picture is Derby -- my daughter's cat who drinks from the spigot. Our cat is the grey one posing with the orange wooden cat.

Correctly solved by:

1. Jim Arrison Norristown, Pennsylvania
2. Mike Singer Winchester, Virginia
3. John Funk Ventura, California
4. Walt Arrison Philadelphia, Pennsylvania
5. Alex McIntire Columbus, Georgia
6. William Proffitt Winchester, Virginia
7. Derick Gilkeson Harrisonburg, Virginia
8. James Alarie University of Michigan -- Flint,
Flint, Michigan
9. Charles Washington Winchester, Virginia
10. Arsalan Heydarian Harrisonburg, Virginia
11. Tyler Windham Columbus, Georgia
12. Jason Krech Harrisonburg, Virginia
13. Wajih Ansari Harrisonburg, Virginia
14. António Baptista Lisboa - Portugal
15. Rob Adams Winchester, Virginia
16. Sharina Broughton Old Dominion University,
Norfolk, Virginia
17. Arin Smith Winchester, Virginia
18. Hamza Rashid Harrisonburg, Virginia
19. Jeffrey Gaither Winchester, Virginia
20. Cameron S. Columbus, Georgia
21. David & Judy Dixon Bennettsville, South Carolina
22. Daniel Surber Winchester, Virginia