Arrange the numbers 1-12 at the points of the intersection of the circle below so that the sum of the numbers lying on any given circle is equal to the sum of the numbers lying on any other one.

Under this text are four circles which intersect with each other in many ways. I believe they are the same size and overlap each other equally. I have tried EVERYTHING. I really didn't know of any way to start other than just plugging in numbers randomly into each intersecting area. If you can answer or give a hint, I would appreciate it. If this is not the appropriate place to ask such a question, then please direct me to the appropriate place if you can. I've provided an illustrated version in an attached file. It was created with MS Word. If you don't want to download the file, then you will just have to go by my text explanation of it.

Thank you,
Richard Deal

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My heart goes out to you. Whatever grade you are in I would not offer you such a problem. There's precious little you may learn from it, and the result will not justify the sweat required.

Just one word of advice. Your search must not be entirely random. Note that every number from 1-12 appears on exactly 2 circles. Ask yourself about the sum of numbers on all four circles. It's 4 times the sum on a single circle. On the other hand, it's twice the sum of all numbers from 1 through 12.

This way you get the sum of 6 numbers on a single circle. And I am quite confident no one will offer you more help short of giving away a solution. This is because there's simply nothing add to the above.

When this exercise is over, do not let your teacher off the hook and insist on getting the solution. And I mean that - a solution, not the answer.

Good luck,
Alexander Bogomolny

There's more to be said. The problem is in fact almost trivial and has a multitude of solutions. What you need is one more observation: find that sum of numbers on a single circle. (Just between us it's gong to be 39.) That sum when divided by 3 gives 13 which can be represented as the sum of numbers 1-12 in 6 different ways. So you have 6 pairs of numbers. And 6 pairs of circles. Each pair of circles intersect at exactly two points. Place pairs of numbers on pairs of such intersection points. Any way you do that will work.

Regards,
Alexander Bogomolny