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Magic Heptominoes

presented by Kate Jones
for the attendees of Gathering for Gardner VII
March 16-19, 2006 – Atlanta, Georgia

Magic squares are grids filled with numbers so that every row and column yields the same total. We can do a similar trick with other shapes. For example, there are 108 distinct shapes of 7 squares joined, known as heptominoes. We can fill their cells with the numbers 1 through 7 so that every row of two or more cells contains the same sum.

Aside from the trivial 1x7, there are 32 heptominoes that have the unique capacity to be “magic.” Here they are. Imagine that the 7 individual squares are boxed off, like crossword puzzles, or mark them in yourself. Some are trickier to solve than you’d think. Better use pencil (with eraser), not a pen, to fill them in.

All the heptominoes with magic sums

If they can’t all be the same sums, can they be all different? Among the remaining 75 heptominoes, we can inquire how many can be solved to have all different sums that are also consecutive integers. We find that 16 contain rows of 6-1 and 5-2-2 and thus are not solvable either way. That leaves 59 candidates for a consecutive-sums solution. Five of those are believed to have no solution. The rest are a fine mixed bag of molecular cohesion. Working out their shapes and solutions is left as an adventure for the puzzler. View the whole set.

This series of puzzles is an extension of the “Polyomino Magic” theme developed by Kate Jones for Kadon’s Six Disks set, which includes magic sums on pentominoes and hexominoes and is played with disks numbered 1 through 6.

PRIZE—The first puzzler to identify and send in the five unsolvable heptomino shapes mentioned above (not counting the 16 long, narrow shapes) wins a prize. Send to: Kadon Enterprises, Inc., 1227 Lorene Dr., Suite 16, Pasadena, MD 21122. Or email to Kate Jones.


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