This header plots the critical line of the Riemann Zeta Function.  A complete understanding wins a \$1,000,000 prize.
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Martin Gardner once posed the problem of arranging 20 pennies (or any circular coin) so that each coin touches exactly 3 other coins.  Recently, Tatsuo Kondo found a related solution that involves 14 squares of the same size.  Arrange 14 identical squares so that each square shares a border with exactly 3 other squares. The problem was originally sent to me by Richard Hess.

Solutions were sent to me by Rozberk Omniist, Robert Windshcitl, Erich Friedman, Jukka-Pekka Ikaheimonen, Michael Reid and Joseph DeVincentis.  Erich Friedman found a way to arrange 50 squares that each touch 4 others. Erich has also been studying cubes.  He found 6 cubes that touch 5 others, and 14 cubes that touch 6 others.  He isn't sure that 14 is minimal.  He also believes a configuration where every cube touches 7 others is possible, but hasn't found a configuration.  Michael Reid pointed out that 16 pennies is plenty for every penny to touch 3 others.