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Treebeard's Stumper Answer
19 Dec 97

Another Magic Star

Place any ten different numbers, not necessarily consecutive, in the circles on the Holiday star, so that each row of 4 adds up to 24.

Each row of this holiday star adds up to 24. I tried (and tried) to use just the numbers 1-10. I'm sure it's not possible.

Note: There's an interesting analysis of this problem in answer 393 of 536 Puzzles & Curious Problems by American puzzle master Henry Ernest Dudeney. He notes that after you put any five numbers in the inner pentagon, you can find the numbers on the points for any constant sum by following a rule: to find X, subtract the sum of A and B from half the constant plus C.

For example, 9 = (24 / 2 + 3) - (4 + 2) and 6 = (24 / 2 + 1) - (5 + 2). Some solutions will require repeated numbers, fractions, and even negative numbers. Dudeney also states that in any solution, the constant will be two-fifths of the sum of the ten numbers. It works!

It's easy to find solutions for higher constants, for even numbers at least. Add one to each of the star points, and the sum will be 26. Add two and the sum will be 28. But odd sums are impossible without fractions. And it's not possible with consecutive integers.

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Marc Kummel / mkummel@rain.org