Can You Create Squares of Integral Areas on an 8x8 Grid Using Strings?

[imathgeek]

Hi there,

I am a new person here, so I hope that you can understand this problem as I have written it. Suppose that you have an 8 by 8 grid (like a geo board) where at the intersections of the line segments are posts whereby you may run a string or rubber band about and make all sorts of geometric shapes.

"On the 8 by 8 grid can you form squares with a string or rubberband such that the squares have integral areas from 1 through 9? The lines needn't be horizontal or vertical in order to do this. If possible, how do you form your squares on the grid to achieve the desired area? If not possible, provide a proof showing why it cannot exist."

This is a problem I posed to my geometry students and I have received all sorts of answers. I am looking to verify my own work on the problem. Yep, I am a new professor and gave a problem that I didn't have an answer to.

I know that squares of areas 1, 4 and 9 are trivial. I can place squares with areas 2, 5, and 8. Since these are the only sums of two squares less than 10, these should be the only squares possible.

Any suggestions would be greatly appreciated.

Thanks.

imathgeek

 

[Loren Booda]

The variety of combinations is similar to the myriad "Traveling Salesman" solutions. Go Google on "traveling salesman".

 

[imathgeek]

I guess that I could look at the certain discrete values of the perimeter if that is what you're implying.

Thanks for the assistance. After reading your many posts this afternoon, I had a feeling that you would have something constructive to add to the problem.

Ken

 

[Hurkyl]

To form a square of area A you need a side of sqrt(A)

To form a side of sqrt(A), it must be the distance between two lattice points, so there are integers B and C with A = B^2 + C^2, so your hypothesis is correct.

Hurkyl