What is the area of the largest square that can fit entirely within a unit cube?

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# A plane old stuffed cube

Best Answer plasmid, 27 September 2013 - 04:49 AM

If no one else is going to go after this one... I can give an answer based on, well, working in the spirit of the best solution the program could find. But I can't prove that there aren't any larger squares that can fit in the unit cube.

### #1

Posted 24 September 2013 - 02:44 AM

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #2

Posted 24 September 2013 - 03:25 AM

I just realized where I went wrong in my first guess... I'll have to figure a different answer:

**Edited by Dariusray, 24 September 2013 - 03:34 AM.**

### #3

Posted 24 September 2013 - 03:26 AM

### #4

Posted 24 September 2013 - 03:37 AM

I just realized where I went wrong in my first guess... I'll have to figure a different answer:

Spoiler for Estimation now...

Since it's a unit cube, your answer simplifies (if I'm reading it correctly) to Sqrt(2) / 2 = .707.

This is smaller than a cube face. But maybe I don't interpret your answer correctly.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #5

Posted 24 September 2013 - 03:39 AM

Spoiler for First guesses

A little too large, but close. And the answer is in fact a rational number.

If it helps, there is a close relation to the question that asks whether a cube can be pushed through a square hole in a smaller cube.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #6

Posted 25 September 2013 - 03:42 AM

So far the best that my horribly inefficient and possibly buggy java code came up with is an area of 1.0984. But it's still running.

I probably ought to have searched for a reasonably efficient Windows C compiler instead.

Unfortunately I don't see any easy way of modifying the code to work in 4 dimensions. Especially since it uses cross products which I don't think are defined in 4D.

### #7

Posted 26 September 2013 - 02:42 AM

This is what my brute force approach came up with. If you look at the coordinates of the square that it fit into the unit cube, it should become clear how to imagine that it's oriented, and allow you to come up with a more analytical approach to solving the problem. Which I'm not going to attempt myself for now.

### #8

Posted 26 September 2013 - 07:23 AM

I'm learning how to do java programming.

You get an area of 1.0984, a rational number.

But it can be bigger than that.

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #9

Posted 27 September 2013 - 04:49 AM Best Answer

### #10

Posted 11 October 2013 - 08:47 PM

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

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