View Full Version : A little geometry (topology?) puzzler

Rich Mann
04-28-2006, 08:43 PM
I am thinking about an art project that, in part, consists of a large square divided into smaller square areas. It occurred to me that it would be more elegant (in the sense of an elegant solution or proof) if every sub-square was a different size--and more elegant still if the subsquares were inclusive starting with one.

I began adding up the squares of the whole numbers, in order, and found that the cumulative sum of the first 24 squares added to 4900 (70 squared). I don't know if this, necessarily, means that those 24 smaller squares can be jigsawed into a 70x70 area, or not. It seemed to me, though, that this is the kind of thing that someone might have figured out before.

So, before I cut out a bunch of little squares of paper and start fiddling with them or try to write a complicated, brute-force program, I thought I might as well ask the smartest and most knowledgeable people that I know about it.

So, do any 'Dopers out there know if there is a solution to this puzzle and can point me to it?

04-28-2006, 08:53 PM
Well according to this site (http://home.tiscalinet.ch/t_wolf/tw/misc/squares.html) there is no solution. There are a few really good tries, though.

Funny thing is I thought I remembered this from an old book of Sam Lloyd's puzzles and would have said that it had a solution.

04-28-2006, 08:58 PM
I'm guessing that what you are describing is called "squaring the square". That is you want to get a square and divide it into smaller squares, all of which are of different sizes?
From what I understand from recreational mathematics this was VERY difficult to achieve and the first success (but not the simplest solution) was finally found in 1939.
You can read more on the subject here:

04-28-2006, 08:59 PM
Of course, if you're not hung up on using squares, it's very easy to tile with rectangles.

Rich Mann
04-28-2006, 09:03 PM
Thanks MonkeyMensch! That's a useful and interesting site. A bit disheartening, though.

It's a good thing I checked here before wasting my time. That's what I love about being a part of this place!

Rich Mann
04-28-2006, 09:27 PM
Yeah, wolf_meister, that's more or less the idea.

I really would have liked to have been able to use the 70x70 square, but the solution given in the article you linked will do well enough.

Thanks for the help.

04-28-2006, 10:19 PM
At first I thought the other Dopers had the answer you wanted but I'm glad my information was also helpful to you.

At least you have a solution although the squares are not consecutive integers.

04-29-2006, 02:20 AM
It turns out that 4900 is the ONLY number with the property you describe. That is, it is the only square number that is also expressible as 1^2 + 2^2 + 3^2 + ... + n^2, except for the trivial case of 1=1^2=1^2. For proof of this fact, google the "cannonball problem".