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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.