# Proposing a Recreational math question/problem

• imathgeek
In summary, the conversation discusses a problem involving forming squares with integral areas from 1 through 9 on an 8 by 8 grid using a string or rubber band. The conversation includes different possible solutions and approaches to the problem, as well as a reference to the "Traveling Salesman" problem and the use of lattice points to form sides of a square. The conclusion is that it is possible to form squares with areas 1, 4, and 9, as well as 2, 5, and 8, but not with any other areas due to the limitations of lattice points.

#### 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

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

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

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

## 1. What is recreational math?

Recreational math is the branch of mathematics that focuses on solving mathematical problems and puzzles for the purpose of entertainment and enjoyment, rather than practical applications.

## 2. How do I come up with a good recreational math question/problem?

A good recreational math question/problem should be challenging, but also have a clear and logical solution. Start by playing with numbers and patterns, and see if you can create a problem that requires creative thinking and problem-solving skills.

## 3. Is recreational math only for advanced mathematicians?

No, recreational math is for anyone who enjoys solving puzzles and playing with numbers. While some problems may require advanced mathematical concepts, many can be solved with basic math skills and creative thinking.

## 4. Can recreational math have real-life applications?

Yes, although recreational math is primarily done for fun, many problems and puzzles have practical applications in fields such as computer science, physics, and engineering. Additionally, solving recreational math problems can improve critical thinking skills and problem-solving abilities.

## 5. Where can I find resources for recreational math problems?

There are many websites, books, and online communities dedicated to recreational math. Some popular resources include websites like Project Euler and Brilliant, as well as books by authors such as Martin Gardner and Ian Stewart. You can also find many interesting problems and puzzles by searching online or participating in online forums and discussions.

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