Cat Puzzle

[Joseph Bolas]

Here is a maths puzzle taken from a PerplexCity card. Hopefully it ain't too easy.

The cat herders of Harbin have a long tradition of animal husbandry. But every year, they suffer a certain amount of natural wastage as cats stray from the fold. Over the centuries, these mathematically minded herders have calculated the minimum size for a profitable herd. See if you can figure it out from the herders' poem:

In Harbin, the cats, I declare,
They numbered one third of a square,
If a quarter then strayed,
Still a cube will have stayed,
How many, at least, must there be?

[Paul Howe]

972, I think.

You need to work through the list of cubes, multiply each one by 4, check its a square number and if so divide by 3 to get the answer. 9^3 = 729 is the first cube this works for (4*4^3 = 256 is a square too, but doesn't give an integer when dividing by 3), and 729*3/4 = 972

[Ben Wilson]

972, I think.

You need to work through the list of cubes, multiply each one by 4, check its a square number and if so divide by 3 to get the answer. 9^3 = 729 is the first cube this works for (4*4^3 = 256 is a square too, but doesn't give an integer when dividing by 3).

1 also works but fails the second condition.

[Paul Howe]

972, I think.

You need to work through the list of cubes, multiply each one by 4, check its a square number and if so divide by 3 to get the answer. 9^3 = 729 is the first cube this works for (4*4^3 = 256 is a square too, but doesn't give an integer when dividing by 3).

1 also works but fails the second condition.

True, but you could hardly call that a herd .

[Ben Wilson]

972, I think.

You need to work through the list of cubes, multiply each one by 4, check its a square number and if so divide by 3 to get the answer. 9^3 = 729 is the first cube this works for (4*4^3 = 256 is a square too, but doesn't give an integer when dividing by 3).

1 also works but fails the second condition.

True, but you could hardly call that a herd .

They may be several very unfortunate amputees.

[Conor]

Yeah. I get 729 too (after some initial errors :p). You can make the checking process quicker by only cubing values divisible by 3.

[Paul Howe]

Yeah. I get 729 too (after some initial errors :p). You can make the checking process quicker by only cubing values divisible by 3.

Very true. Its even quicker to ask MATLAB though

[Joseph Bolas]

Well done all, the answer is 972.

The way I solved this was to first off, find cubes divisible by 3 and then add one third to each of these (to give the number of cats, before 1 quarter strayed). I then multipled these numbers by 3 and worked out which one was a square.

It would take you just 3 cubes (27, 216 and 729), before you find the answer:

729 / 3 = 243,
729 + 243 = 972,
972 x 3 = 2916
√2916 = 54

Therefore the number of cats was 972 .