# 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?

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

 200 (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 4/1/16.