# Problem of the Month (June 2013)

One of the problems in the November 2003 Math Magic was to investigate Hanoi square tilings, tilings using integer-sided squares with each square resting on a larger square, and at least 2 squares in the bottom row. But that problem was restricted to tiling squares. This month we want to know how high rectangles can be tiled for n≤100.

We can ask the same question about cubes. What are the highest Hanoi boxes? For which n≤100 is the highest box higher than just the sum of the proper factors of n?

Highest-Known Hanoi Towers
 height = 1 height = 1 height = 3 height = 3 height = 6 height = 4 height = 7 height = 6 height = 11 height = 8 height = 16 height = 11 height = 14 height = 12 height = 19 height = 15 height = 22 height = 15 height = 25 height = 21 height = 27 height = 21 height = 36 height = 26 height = 32 height = 27 height = 36(M. Morandi) height = 30 height = 42(Bryce Herdt) height = 31 height = 42 height = 35(M. Morandi) height = 44 height = 37 height = 57(M. Morandi)
 height = 40 height = 50(Maurizio Morandi) height = 42(Maurizio Morandi) height = 55 height = 46(Maurizio Morandi) height = 62(Maurizio Morandi) height = 46 height = 60(Maurizio Morandi) height = 51 height = 62 height = 50(Maurizio Morandi) height = 77 height = 55(Maurizio Morandi) height = 69(Maurizio Morandi) height = 57(Maurizio Morandi) height = 71(Maurizio Morandi) height = 62(Maurizio Morandi) height = 81 height = 72(Maurizio Morandi) height = 85(Maurizio Morandi) height = 71(Maurizio Morandi) height = 83(Maurizio Morandi) height = 72(Maurizio Morandi) height = 111(Maurizio Morandi) height = 69(Bryce Herdt) height = 94(Joe DeVincentis) height = 78(Maurizio Morandi) height = 90(Joe DeVincentis) height = 78(Maurizio Morandi) height = 104(Maurizio Morandi) height = 81(Maurizio Morandi) height = 97(Joe DeVincentis) height = 81(Maurizio Morandi) height = 104(Maurizio Morandi) height = 87(Joe DeVincentis) height = 124(Joe DeVincentis)
 height = 88(Maurizio Morandi) height = 108(Maurizio Morandi) height = 87(Maurizio Morandi) height = 113(Maurizio Morandi) height = 89(Maurizio Morandi) height = 125(Maurizio Morandi) height = 94(Joe DeVincentis) height = 117(Joe DeVincentis) height = 102(Maurizio Morandi) height = 124(Joe DeVincentis) height = 104(Maurizio Morandi) height = 151(Joe DeVincentis) height = 109(Maurizio Morandi) height = 128(Joe DeVincentis) height = 105(Maurizio Morandi) height = 146(Joe DeVincentis) height = 113(Maurizio Morandi) height = 150(Joe DeVincentis) height = 110(Maurizio Morandi) height = 136(Maurizio Morandi) height = 114(Maurizio Morandi) height = 144(Maurizio Morandi) height = 121(Maurizio Morandi) height = 164(Maurizio Morandi) height = 126(Joe DeVincentis) height = 145(Maurizio Morandi) height = 121(Maurizio Morandi) height = 153(Maurizio Morandi)

What are the thinnest Hanoi towers that contain all the squares of sides 1 through n? Here are the best-known results:

Thinnest-Known Hanoi Towers Containing 1 Through n
 width = 2 width = 4 width = 5 width = 8 width = 10 width = 16 width = 20 width = 22(Bryce Herdt) width = 31(Bryce Herdt) width = 37 width = 41(Maurizio Morandi) width = 45(Maurizio Morandi) width = 57 width = 64 width = 67 width = 77(Bryce Herdt) width = 85(Joe DeVincentis)

Here are the important layers for the tallest known Hanoi boxes that are taller than the sum of the proper divisors of n:

 n=22height = 1811+6+111+5+211+4+2+1 n=26height = 2013+713+6+113+4+2+1 n=33height = 2011+911+6+311+5+3+1 n=34height = 2417+6+117+5+217+4+2+1 n=38height = 2919+8+219+7+2+119+6+3+1 n=39height = 2313+9+113+7+313+6+3+1 n=43height = 887+16+25+3(Joe DeVincentis) n=44height = 4822+11+10+522+11+8+4+2+122+11+6+4+3+2(Joe DeVincentis) n=46height = 3523+10+223+8+423+6+3+2+1(Joe DeVincentis) n=50height = 4525+10+8+225+10+7+2+125+10+6+3+125+10+5+4+1(Joe DeVincentis) n=51height = 2917+11+117+9+317+6+3+2+1(Joe DeVincentis) n=52height = 5426+13+12+326+13+10+526+13+8+4+2+1(Maurizio Morandi) n=57height = 3119+11+119+7+519+8+419+6+3+2+1(Joe DeVincentis) n=58height = 4429+10+529+9+629+8+4+2+1(Joe DeVincentis) n=59height = 887+15+3(Joe DeVincentis)
 n=62height = 4631+13+231+12+331+10+531+9+631+8+4+2+1(Joe DeVincentis) n=64height = 6432+16+8+7+132+16+8+6+232+16+8+5+2+132+16+8+4+3+1(Joe DeVincentis) n=65height = 2413+1113+10+113+7+3+113+6+3+2(Joe DeVincentis) n=66height = 8333+22+18+9+133+22+15+10+333+22+12+6+4+3+2+1(Joe DeVincentis) n=68height = 6934+17+12+4+234+17+12+3+2+134+17+10+5+2+134+17+8+4+3+2+1(top view below)(Bryce Herdt) n=69height = 3823+12+323+12+2+123+11+3+123+9+3+2+1(top view below)(Bryce Herdt) n=71height = 1312+110+29+3+18+4+17+5+1(Joe DeVincentis) n=73height = 1211+110+29+36+3+2+1(Joe DeVincentis) n=74height = 5237+13+237+10+537+9+637+8+4+2+1(Joe DeVincentis) n=75height = 5325+15+1325+15+12+125+15+9+3+1(Joe DeVincentis) n=76height = 8538+19+16+8+438+19+14+7+4+2+138+19+12+6+4+3+2+1(Joe DeVincentis) n=78height = 9939+26+13+12+6+339+26+13+10+6+539+26+13+9+6+3+2+1(Joe DeVincentis) n=79height = 13139+3+18+4+1(Maurizio Morandi) n=81height = 4427+9+7+127+9+6+227+9+5+3(Joe DeVincentis) n=82height = 6541+18+641+(24)41+16+841+15+6+341+12+6+4+2(Joe DeVincentis) n=83height = 887+16+25+35+2+1(Joe DeVincentis) n=85height = 3317+15+117+11+517+10+5+1(Joe DeVincentis) n=86height = 6043+1743+16+143+14+2+143+10+5+2(Joe DeVincentis) n=87height = 4129+11+129+9+329+6+3+2+1(Joe DeVincentis) n=88height = 9744+22+20+10+144+22+16+8+4+2+144+22+12+8+6+4+1(Joe DeVincentis) n=89height = 121211+19+38+4(Joe DeVincentis) n=91height = 2513+11+113+9+313+8+413+7+4+1(Joe DeVincentis) n=92height = 9746+23+20+5+2+146+23+16+8+446+23+12+6+4+3+2+1(Joe DeVincentis) n=93height = 4631+12+331+12+2+131+9+3+2+131+11+3+1(top view below)(Joe DeVincentis) n=94height = 7147+19+547+19+4+147+19+3+247+16+847+14+7+2+147+12+6+4+2(Joe DeVincentis) n=95height = 3119+11+119+10+219+9+319+8+4(Joe DeVincentis) n=97height = 1413+112+29+58+4+2(Joe DeVincentis) n=98height = 7849+14+10+549+14+9+649+14+8+4+2+1(Joe DeVincentis) n=100height = 11950+25+20+10+8+4+250+25+20+10+7+4+2+150+25+20+10+6+4+3+1(Joe DeVincentis)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/27/13.