c4countdown

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?

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

Paul Howe wrote: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.

Ben Wilson wrote:

Paul Howe wrote: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 .

Paul Howe wrote:

Ben Wilson wrote:

Paul Howe wrote: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.

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

Conor wrote: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

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 .