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

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=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.