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        riddles >> hard >> 21 squares (solution)
        
(Message started by: tim on Jul 28^th, 2002, 12:40am)

Title: 21 squares (solution)
Post by tim on Jul 28^th, 2002, 12:40am

I was attracted to this one because it claimed "calculator/computer power". Them's fightin' words :)

It looked like one that I could do with pencil and paper at worst. And it turned out to be so.

My first approach was to look at the smallest squares, and guess that they had small integer values. First I tried n=1, and guessed s=2n and i=3n visually. Each internal line gave equations for the sum of sizes of squares on either side. Solving a few of these led to o=3.5, so I bumped n up to 2.

With a combination of eyeballing relative sizes (i.e. guessing) and a little equation-solving, I managed to find a set of sizes that satisfied all the equations. As it turned out, the first four relative sizes I guesses were right.

If that hadn't worked, I expect I would have written down all the linear equations and solved them by brute force :)

Enough rambling. The solution:
a=50, b=35, c=27, d=8, e=19, f=15, g=17, h=11, i=6, j=24, k=29, l=25, m=9, n=2, o=7, p=18, q=16, r=42, s=4, t=37, u=33.

Title: Re: 21 squares (solution)
Post by Nicodemus on Jul 28^th, 2002, 2:44pm

That works, Tim. I think that's how it's supposed to be done. But this also occurred to me:

It's designated "CPU" so we can use a computer to solve it. The problem is presented as a digital image... composed of discrete pixels. Load it into photoshop and measure regions in pixels. :)

Title: Re: 21 squares (solution)
Post by tim on Jul 28^th, 2002, 3:35pm

Unfortunately that doesn't work: the image is 216x216 pixels, while the only integer solutions are multiples of 112 pixels. :/

If you actually carried out this procedure, you would be tripped up by the blurry edges of the squares. You would have to decide for each square whether it was really x pixels or x+1 pixels in size, and would find that neither choice worked in the big picture.

That might lead you to erroneously conclude that the problem had no solution.

Nice try, though ;)