Problem of the Month (April 2017)

Given three polyominoes of the same area, what is the smallest rectangle they can tile so that no congruent polyominoes touch along an edge? What if we wish to tile rectangles tilted at various angles?

ANSWERS

Tetromino Rectangles
16
20
24
36

Pentomino Rectangles
15

20

25

30

35

40

45

60

80

135

160

Hexomino Rectangles
18

24

30

36

42

48

54

60

66

72

84

90

96

120

126

150
(Patrick Hamlyn)

168
(Patrick Hamlyn)
(Patrick Hamlyn)

240
(Patrick Hamlyn)

Heptomino Rectangles
21
(Patrick Hamlyn)

28
(Patrick Hamlyn)

(Patrick Hamlyn)

35
(Patrick Hamlyn)

42
(Patrick Hamlyn)
(Patrick Hamlyn)

49
(Patrick Hamlyn)

56
(Patrick Hamlyn)

(Patrick Hamlyn)

63
(Patrick Hamlyn)

70
(Patrick Hamlyn)
(Patrick Hamlyn)

77
(Patrick Hamlyn)

84
(Patrick Hamlyn)
(Patrick Hamlyn)
(Patrick Hamlyn)

112
(Patrick Hamlyn)

(Patrick Hamlyn)

140
(Patrick Hamlyn)

168
(Patrick Hamlyn)

224
(Patrick Hamlyn)

Octomino Rectangles
24
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

32
(Patrick Hamlyn)

(Patrick Hamlyn)

40
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

48
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

56
(Patrick Hamlyn)

(Patrick Hamlyn)

64
(Patrick Hamlyn)

(Patrick Hamlyn)

72
(Patrick Hamlyn)

(Patrick Hamlyn)

(Patrick Hamlyn)

80
(Patrick Hamlyn)
(Patrick Hamlyn)

(Patrick Hamlyn)

88
(Patrick Hamlyn)

96
(Patrick Hamlyn)

(Patrick Hamlyn)

112
(Patrick Hamlyn)
(Patrick Hamlyn)
(Patrick Hamlyn)

120
(Patrick Hamlyn)

128
(Patrick Hamlyn)

144
(Patrick Hamlyn)

160
(Patrick Hamlyn)
(Patrick Hamlyn)

192
(Patrick Hamlyn)

200
(Patrick Hamlyn)

240
(Patrick Hamlyn)

Tetromino Diamonds
32

Pentomino Diamonds
25

50

60

85

140

160
(George Sicherman)

200
(George Sicherman)

Hexomino Diamonds
18

48

60

72

Heptomino Diamonds
28
(George Sicherman)

63
(George Sicherman)

77
(George Sicherman)

98
(George Sicherman)

140
(George Sicherman)

182
(George Sicherman)

196
(George Sicherman)

Pentomino Slope=2 Rectangles
35

40

130

180
(George Sicherman)

225
(George Sicherman)

260
(George Sicherman)

Hexomino Slope=2 Rectangles
54

96

300
(George Sicherman)
(George Sicherman)

Heptomino Slope=2 Rectangles
35

Tetromino Slope=3 Rectangles
56
(George Sicherman)
(George Sicherman)

Pentomino Slope=3 Rectangles
75
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

85
(George Sicherman)
(George Sicherman)
(George Sicherman)

170
(George Sicherman)
(George Sicherman)

230
(George Sicherman)

270
(George Sicherman)

290
(George Sicherman)

Hexomino Slope=3 Rectangles
18
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

36
(George Sicherman)
(George Sicherman)
(George Sicherman)
(George Sicherman)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/17.