24 Coins

  1. You have 24 equally round coins. Can you put them on a table (each coin lying flat on the table) such that each coin touches exactly three other coins?
  2. The same question for 25 coins.
  3. Can you generalize this? For what numbers of coins can this be done?

Source: Pertsel Vladimir, c/o internet newsgroup rec.puzzles. Used as Ken's POTD 6/1/94.


Solutions were received from Erich Friedman, Larry Baum, and Philippe Fondanaiche.

All solvers showed that there are solutions for any even number of coins (>=16). There are no solutions for an odd number of coins: Let n be the # of coins. Then each coin touches 3 others, so there are 3n/2 touches, since each touch is counted twice. Hence n must be even.
In general, arrange 4 coins into the shape of a diamond, with coins 2 and 3 touching each other and coins 1 and 4 touching 2 and 3 from opposite sides. Several of these diamonds can be arranged in a circle to satisfy any multiple of 4 coins greater than 16. Any group of 4 can be easily be replaced by a group of 6, by putting two coins between 2-3 and 4. For 24 coins, the configuration ends up resembling a hexagon with an additional coin at each corner.
 1      1       a b c d
2-3    2-3     e f   g h
 4     5-6    i         j
        4    k l       m n
              o         p
               q r   s t
                u v w x 


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