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?

ANSWERS

Highest-Known Hanoi Towers

n=2 height = 1	n=3 height = 1	n=4 height = 3	n=5 height = 3	n=6 height = 6	n=7 height = 4	n=8 height = 7
n=9 height = 6	n=10 height = 11	n=11 height = 8	n=12 height = 16	n=13 height = 11	n=14 height = 14	n=15 height = 12
n=16 height = 19	n=17 height = 15	n=18 height = 22	n=19 height = 15	n=20 height = 25	n=21 height = 21	n=22 height = 27
n=23 height = 21	n=24 height = 36	n=25 height = 26	n=26 height = 32	n=27 height = 27	n=28 height = 36 (M. Morandi)	n=29 height = 30
n=30 height = 42 (Bryce Herdt)	n=31 height = 31	n=32 height = 42	n=33 height = 35 (M. Morandi)	n=34 height = 44	n=35 height = 37	n=36 height = 57 (M. Morandi)

n=37 height = 40	n=38 height = 50 (Maurizio Morandi)	n=39 height = 42 (Maurizio Morandi)	n=40 height = 55	n=41 height = 46 (Maurizio Morandi)
n=42 height = 62 (Maurizio Morandi)	n=43 height = 46	n=44 height = 60 (Maurizio Morandi)	n=45 height = 51	n=46 height = 62
n=47 height = 50 (Maurizio Morandi)	n=48 height = 77	n=49 height = 55 (Maurizio Morandi)	n=50 height = 69 (Maurizio Morandi)	n=51 height = 57 (Maurizio Morandi)
n=52 height = 71 (Maurizio Morandi)	n=53 height = 62 (Maurizio Morandi)	n=54 height = 81	n=55 height = 72 (Maurizio Morandi)	n=56 height = 85 (Maurizio Morandi)
n=57 height = 71 (Maurizio Morandi)	n=58 height = 83 (Maurizio Morandi)	n=59 height = 72 (Maurizio Morandi)	n=60 height = 111 (Maurizio Morandi)	n=61 height = 69 (Bryce Herdt)
n=62 height = 94 (Joe DeVincentis)	n=63 height = 78 (Maurizio Morandi)	n=64 height = 90 (Joe DeVincentis)	n=65 height = 78 (Maurizio Morandi)	n=66 height = 104 (Maurizio Morandi)
n=67 height = 81 (Maurizio Morandi)	n=68 height = 97 (Joe DeVincentis)	n=69 height = 81 (Maurizio Morandi)	n=70 height = 104 (Maurizio Morandi)	n=71 height = 87 (Joe DeVincentis)

n=72 height = 124 (Joe DeVincentis)	n=73 height = 88 (Maurizio Morandi)	n=74 height = 108 (Maurizio Morandi)	n=75 height = 87 (Maurizio Morandi)
n=76 height = 113 (Maurizio Morandi)	n=77 height = 89 (Maurizio Morandi)	n=78 height = 125 (Maurizio Morandi)	n=79 height = 94 (Joe DeVincentis)
n=80 height = 117 (Joe DeVincentis)	n=81 height = 102 (Maurizio Morandi)	n=82 height = 124 (Joe DeVincentis)	n=83 height = 104 (Maurizio Morandi)
n=84 height = 151 (Joe DeVincentis)	n=85 height = 109 (Maurizio Morandi)	n=86 height = 128 (Joe DeVincentis)	n=87 height = 105 (Maurizio Morandi)
n=88 height = 146 (Joe DeVincentis)	n=89 height = 113 (Maurizio Morandi)	n=90 height = 150 (Joe DeVincentis)	n=91 height = 110 (Maurizio Morandi)
n=92 height = 136 (Maurizio Morandi)	n=93 height = 114 (Maurizio Morandi)	n=94 height = 144 (Maurizio Morandi)	n=95 height = 121 (Maurizio Morandi)
n=96 height = 164 (Maurizio Morandi)	n=97 height = 126 (Joe DeVincentis)	n=98 height = 145 (Maurizio Morandi)	n=99 height = 121 (Maurizio Morandi)
n=100 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=22
height = 18

11+6+1
11+5+2
11+4+2+1 n=26
height = 20

13+7
13+6+1
13+4+2+1 n=33
height = 20

11+9
11+6+3
11+5+3+1 n=34
height = 24

17+6+1
17+5+2
17+4+2+1 n=38
height = 29

19+8+2
19+7+2+1
19+6+3+1 n=39
height = 23

13+9+1
13+7+3
13+6+3+1 n=43
height = 8

8
7+1
6+2
5+3
(Joe DeVincentis) n=44
height = 48

22+11+10+5
22+11+8+4+2+1
22+11+6+4+3+2
(Joe DeVincentis)
n=46
height = 35

23+10+2
23+8+4
23+6+3+2+1
(Joe DeVincentis) n=50
height = 45

25+10+8+2
25+10+7+2+1
25+10+6+3+1
25+10+5+4+1
(Joe DeVincentis) n=51
height = 29

17+11+1
17+9+3
17+6+3+2+1
(Joe DeVincentis) n=52
height = 54

26+13+12+3
26+13+10+5
26+13+8+4+2+1
(Maurizio Morandi) n=57
height = 31

19+11+1
19+7+5
19+8+4
19+6+3+2+1
(Joe DeVincentis) n=58
height = 44

29+10+5
29+9+6
29+8+4+2+1
(Joe DeVincentis) n=59
height = 8

8
7+1
5+3
(Joe DeVincentis)

n=62
height = 46

31+13+2
31+12+3
31+10+5
31+9+6
31+8+4+2+1
(Joe DeVincentis) n=64
height = 64

32+16+8+7+1
32+16+8+6+2
32+16+8+5+2+1
32+16+8+4+3+1
(Joe DeVincentis) n=65
height = 24

13+11
13+10+1
13+7+3+1
13+6+3+2
(Joe DeVincentis) n=66
height = 83

33+22+18+9+1
33+22+15+10+3
33+22+12+6+4+3+2+1
(Joe DeVincentis) n=68
height = 69

34+17+12+4+2
34+17+12+3+2+1
34+17+10+5+2+1
34+17+8+4+3+2+1
(top view below)
(Bryce Herdt) n=69
height = 38

23+12+3
23+12+2+1
23+11+3+1
23+9+3+2+1
(top view below)
(Bryce Herdt)
n=71
height = 13

12+1
10+2
9+3+1
8+4+1
7+5+1
(Joe DeVincentis) n=73
height = 12

11+1
10+2
9+3
6+3+2+1
(Joe DeVincentis) n=74
height = 52

37+13+2
37+10+5
37+9+6
37+8+4+2+1
(Joe DeVincentis) n=75
height = 53

25+15+13
25+15+12+1
25+15+9+3+1
(Joe DeVincentis) n=76
height = 85

38+19+16+8+4
38+19+14+7+4+2+1
38+19+12+6+4+3+2+1
(Joe DeVincentis) n=78
height = 99

39+26+13+12+6+3
39+26+13+10+6+5
39+26+13+9+6+3+2+1
(Joe DeVincentis)
n=79
height = 13

13
9+3+1
8+4+1
(Maurizio Morandi) n=81
height = 44

27+9+7+1
27+9+6+2
27+9+5+3
(Joe DeVincentis) n=82
height = 65

41+18+6
41+(24)
41+16+8
41+15+6+3
41+12+6+4+2
(Joe DeVincentis) n=83
height = 8

8
7+1
6+2
5+3
5+2+1
(Joe DeVincentis) n=85
height = 33

17+15+1
17+11+5
17+10+5+1
(Joe DeVincentis) n=86
height = 60

43+17
43+16+1
43+14+2+1
43+10+5+2
(Joe DeVincentis)
n=87
height = 41

29+11+1
29+9+3
29+6+3+2+1
(Joe DeVincentis) n=88
height = 97

44+22+20+10+1
44+22+16+8+4+2+1
44+22+12+8+6+4+1
(Joe DeVincentis) n=89
height = 12

12
11+1
9+3
8+4
(Joe DeVincentis) n=91
height = 25

13+11+1
13+9+3
13+8+4
13+7+4+1
(Joe DeVincentis) n=92
height = 97

46+23+20+5+2+1
46+23+16+8+4
46+23+12+6+4+3+2+1
(Joe DeVincentis) n=93
height = 46

31+12+3
31+12+2+1
31+9+3+2+1
31+11+3+1
(top view below)
(Joe DeVincentis)
n=94
height = 71

47+19+5
47+19+4+1
47+19+3+2
47+16+8
47+14+7+2+1
47+12+6+4+2
(Joe DeVincentis) n=95
height = 31

19+11+1
19+10+2
19+9+3
19+8+4
(Joe DeVincentis) n=97
height = 14

13+1
12+2
9+5
8+4+2
(Joe DeVincentis) n=98
height = 78

49+14+10+5
49+14+9+6
49+14+8+4+2+1
(Joe DeVincentis) n=100
height = 119

50+25+20+10+8+4+2
50+25+20+10+7+4+2+1
50+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.

n=22 height = 18 11+6+1 11+5+2 11+4+2+1	n=26 height = 20 13+7 13+6+1 13+4+2+1	n=33 height = 20 11+9 11+6+3 11+5+3+1	n=34 height = 24 17+6+1 17+5+2 17+4+2+1	n=38 height = 29 19+8+2 19+7+2+1 19+6+3+1	n=39 height = 23 13+9+1 13+7+3 13+6+3+1	n=43 height = 8 8 7+1 6+2 5+3 (Joe DeVincentis)	n=44 height = 48 22+11+10+5 22+11+8+4+2+1 22+11+6+4+3+2 (Joe DeVincentis)
n=46 height = 35 23+10+2 23+8+4 23+6+3+2+1 (Joe DeVincentis)	n=50 height = 45 25+10+8+2 25+10+7+2+1 25+10+6+3+1 25+10+5+4+1 (Joe DeVincentis)	n=51 height = 29 17+11+1 17+9+3 17+6+3+2+1 (Joe DeVincentis)	n=52 height = 54 26+13+12+3 26+13+10+5 26+13+8+4+2+1 (Maurizio Morandi)	n=57 height = 31 19+11+1 19+7+5 19+8+4 19+6+3+2+1 (Joe DeVincentis)	n=58 height = 44 29+10+5 29+9+6 29+8+4+2+1 (Joe DeVincentis)	n=59 height = 8 8 7+1 5+3 (Joe DeVincentis)

n=62 height = 46 31+13+2 31+12+3 31+10+5 31+9+6 31+8+4+2+1 (Joe DeVincentis)	n=64 height = 64 32+16+8+7+1 32+16+8+6+2 32+16+8+5+2+1 32+16+8+4+3+1 (Joe DeVincentis)	n=65 height = 24 13+11 13+10+1 13+7+3+1 13+6+3+2 (Joe DeVincentis)	n=66 height = 83 33+22+18+9+1 33+22+15+10+3 33+22+12+6+4+3+2+1 (Joe DeVincentis)	n=68 height = 69 34+17+12+4+2 34+17+12+3+2+1 34+17+10+5+2+1 34+17+8+4+3+2+1 (top view below) (Bryce Herdt)	n=69 height = 38 23+12+3 23+12+2+1 23+11+3+1 23+9+3+2+1 (top view below) (Bryce Herdt)
n=71 height = 13 12+1 10+2 9+3+1 8+4+1 7+5+1 (Joe DeVincentis)	n=73 height = 12 11+1 10+2 9+3 6+3+2+1 (Joe DeVincentis)	n=74 height = 52 37+13+2 37+10+5 37+9+6 37+8+4+2+1 (Joe DeVincentis)	n=75 height = 53 25+15+13 25+15+12+1 25+15+9+3+1 (Joe DeVincentis)	n=76 height = 85 38+19+16+8+4 38+19+14+7+4+2+1 38+19+12+6+4+3+2+1 (Joe DeVincentis)	n=78 height = 99 39+26+13+12+6+3 39+26+13+10+6+5 39+26+13+9+6+3+2+1 (Joe DeVincentis)
n=79 height = 13 13 9+3+1 8+4+1 (Maurizio Morandi)	n=81 height = 44 27+9+7+1 27+9+6+2 27+9+5+3 (Joe DeVincentis)	n=82 height = 65 41+18+6 41+(24) 41+16+8 41+15+6+3 41+12+6+4+2 (Joe DeVincentis)	n=83 height = 8 8 7+1 6+2 5+3 5+2+1 (Joe DeVincentis)	n=85 height = 33 17+15+1 17+11+5 17+10+5+1 (Joe DeVincentis)	n=86 height = 60 43+17 43+16+1 43+14+2+1 43+10+5+2 (Joe DeVincentis)
n=87 height = 41 29+11+1 29+9+3 29+6+3+2+1 (Joe DeVincentis)	n=88 height = 97 44+22+20+10+1 44+22+16+8+4+2+1 44+22+12+8+6+4+1 (Joe DeVincentis)	n=89 height = 12 12 11+1 9+3 8+4 (Joe DeVincentis)	n=91 height = 25 13+11+1 13+9+3 13+8+4 13+7+4+1 (Joe DeVincentis)	n=92 height = 97 46+23+20+5+2+1 46+23+16+8+4 46+23+12+6+4+3+2+1 (Joe DeVincentis)	n=93 height = 46 31+12+3 31+12+2+1 31+9+3+2+1 31+11+3+1 (top view below) (Joe DeVincentis)
n=94 height = 71 47+19+5 47+19+4+1 47+19+3+2 47+16+8 47+14+7+2+1 47+12+6+4+2 (Joe DeVincentis)	n=95 height = 31 19+11+1 19+10+2 19+9+3 19+8+4 (Joe DeVincentis)	n=97 height = 14 13+1 12+2 9+5 8+4+2 (Joe DeVincentis)	n=98 height = 78 49+14+10+5 49+14+9+6 49+14+8+4+2+1 (Joe DeVincentis)	n=100 height = 119 50+25+20+10+8+4+2 50+25+20+10+7+4+2+1 50+25+20+10+6+4+3+1 (Joe DeVincentis)