Recreational Mathematics
 

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?

--snip--
AGREED!

Yes.

Yeah, but apart from us.

5!

I've heard warm milk works.

Zen, you are so hot right now.

Here are two: 1 and i
The fallacy for i is that it depends on the order of operations

... like clockwise or counter-clockwise for the hunter and the squirrel.

;)

chilled Vodka works better

I had this dream once and there was like this "planet of numbers"... I said to my self I kept waiting for the signal but it never came.. and then this little oriental guy poped up and said 5-9-10..

:zzz:

n = 10000
;
loop i (1,n)
...loop j (1,n)
......;
......x_square = i^2
......x_cube = j^3
......;
......if x_square = x_cube
.........print i
.........print x_square
......end_if
...end_loop
end_loop
end
store


Results:

x_ = 1
x_square = 1

x_ = 8
x_square = 64

x_ = 27
x_square = 729

x_ = 64
x_square = 4096

x_ = 125
x_square = 15625

x_ = 216
x_square = 46656

x_ = 343
x_square = 117649

x_ = 512
x_square = 262144

x_ = 729
x_square = 531441

x_ = 1000
x_square = 1000000

x_ = 1331
x_square = 1771561

x_ = 1728
x_square = 2985984

x_ = 2197
x_square = 4826809

x_ = 2744
x_square = 7529536

x_ = 3375
x_square = 11390625

x_ = 4096
x_square = 16777216

x_ = 4913
x_square = 24137569

x_ = 5832
x_square = 34012224

x_ = 6859
x_square = 47045881

x_ = 8000
x_square = 64000000

x_ = 9261
x_square = 85766121



You can find the pattern
:p:

Back in the late 50's, when there were only 600 million living Chinese,
my "Can't get to sleep" problem was...

What is the total number of people who have every lived on earth ?

But now, all you have to do is a Google search, and you get this

NUMBER WHO HAVE EVER BEEN BORN 107,602,707,791

Number of people who are worth a shit 150




give or take

Even the title of this thread makes me anxious. I do the maths if I need them, but childhood trauma prevents ever attaching recreation to them.

Counting...