Recreational Mathematics

[ZenGum]

I think we have a full quorum of geeky nerds in here, so this thread might work.

Post your math(s) related ramblings here. Discuss, comment, dispute or drift at your discretion.


I'll go first.


So, a few nights back, unable to sleep, I tried counting sheep. I started getting turned on so I figured something different was needed. I had a little meander through mathematics and although I'm sure it is nothing new or revolutionary, it's mildly interesting so I'll share it here.

I wondered, are there any numbers which are both cubes and squares? In the sense that 16 is square (4x4) and 27 is cubic (3x3x3).

First I thought of 1,000,000 which is 100 cubed and 1,000 squared. Done.

Then I started going through the squares to see if any were cubes. The first I found was 64, being 4 cubed or 8 squared.

Then I noticed that 100 and 1,000 are in a 1:10 ratio, 4 and 8 are 1:2. I wondered, does 1:[any integer] deliver a cubic square?

I tried 3 and 9, but that didn't work (3^3=27, 9^2=81).
Then I tried 9 and 27. 9^3 and 27^2 both = 729.

Ah, so it should work not with 4 and 16, but with 16 and 64. Yup, 4,096.

By this time I saw the general pattern that was going on.
Take a starting number, x. Square it, and let that be the number that gets cubed. Meanwhile, cube x, and let that result be the number which gets squared.

x^2^3 = x^3^2 = cubic square. Also = x ^6, FTR.

Well, of course, now that I see it. Take x, cube it, then square that, and the result will be a cubic square. Well, duh.

It's kind of obvious once you see it, and that is the beauty of this kind of maths. I started with a question I wasn't sure about, and pretty soon, just by reasoning, had not just found an answer (yes) but had understood how these numbers worked, and had a formula for making as many as I want.

Then, because I am actually a liberal arts graduate, I wrote a haiku about this.

Are there cubic squares?
The square of x cubed equals
the cube of x squared.


Final question for discussion: are there any cubic squares other than those described by the x^2^3 formula?