When I tried to stack 21 circles of radii $(30, 31, 32... 50)$ on top of each other in a tube (ID of $100$ wide), I thought they would reach the same height regardless of the order, however I was wrong. Here I tried out three different arrangements of what order to stack the circles, from largest to smallest, alternating largest and smallest, and alternating from middle to largest and working smaller. I cannot come to a clear conclusion of which way would stack all of the circles in the lowest height. Does anyone have any idea of what method of stacking would give the lowest height?

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    $\begingroup$ When the tube was small enough that no two circles would fit next to each other, I proved that the optimal packing was to start with the smallest one, put the next two smaller on either side, and continue, so it would be $50 \dots 38,36,32,30,31,33,35 \dots 49$ I don't remember enough to be sure it works when you can have circles next to each other. $\endgroup$ – Ross Millikan Jan 26 '16 at 5:36
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    $\begingroup$ This is a superb problem, but it definitely deserves a better description. $\endgroup$ – Christian Blatter Jan 26 '16 at 19:16
  • $\begingroup$ I tried to answer a problem which is similar to yours and got an answer not quite the same as yours. FYI: math.stackexchange.com/questions/97842/… $\endgroup$ – Mick Jan 31 '16 at 15:45

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