# Problem of the Month (September 2015)

This month we pack unit squares into small polyominoes. The problems of packing squares into squares, dominoes, and L-triominoes have been studied before. This month we pack squares into straight triominoes and the non-square tetrominoes. What is the smallest of these polyominoes that contains n non-overlapping unit squares?

Solutions were received from Joe DeVincentis and Maurizio Morandi.

Straight Triomino
 3.s = 1 4.s = 4/3 = 1.333+ 5.s = 5/3 = 1.666+ 6.s = 1.913+ (MM) 12.s = 2 14.s = 7/3 = 2.333+ 15.s = (9 + √2)/4 = 2.603+ 16.s = 8/3 = 2.666+ 17.s = 2.799+ 18.s = 2.876+ (JD) 19.s = 2.930+ 20.s = 2.968+ 27.s = 3 30.s = 10/3 = 3.333+ 31.s = 18/5 = 3.600 33.s = 11/3 = 3.666+ 34.s = 3.772+ (JD) 35.s = 3.827+ (JD) 36.s = 3.850+ (JD) 37.s = 3.900+ (JD) 38.s = 3.924+ (JD) 39.s = 3.972+ (JD)

T Tetromino
 1.s = 2√2/3 = .942+ 4.s = 1 5.s = 4/3 = 1.333+ 6.s = 3/2 = 1.500 7.s = 5/3 = 1.666+ 8.s = (4+5√2)/6 = 1.845+ (MM) 9.s = 1.900+ (MM) 16.s = 2 18.s = 7/3 = 2.333+ 19.s = 2 + √2/3 = 2.471+ 20.s = 5/2 = 2.500 21.s = (9 + √2)/4 = 2.603+ 22.s = 8/3 = 2.666+ 23.s = 14/5 = 2.800 (MM) 24.s = (10 + √2)/4 = 2.853+ 25.s = 6(1 + √2)/5 = 2.897+ (MM) 26.s = (12 + 2√2)/5 = 2.965+ (JD) 36.s = 3 39.s = 10/3 = 3.333+ (JD) 40.s = 3.457+ (JD) 42.s = 7/2 = 3.500 (JD) 43.s = (13 + √2)/4 = 3.603+ (JD) 45.s = 11/3 = 3.666+ (JD) 46.s = 3.795+ (JD) 47.s = 1 + 2√2 = 3.828+ (JD) 48.s = 3.853+ (JD) 51.s = 3.972+ (JD)

L Tetromino
 4.s = 1 5.s = 4/3 = 1.333+ 6.s = 3/2 = 1.500 7.s = 5/3 = 1.666+ 8.s = 1.858+ 9.s = 1.931+ (MM) 16.s = 2 18.s = 7/3 = 2.333+ 19.s = 2√3 – 1 = 2.464+ (MM) 20.s = 5/2 = 2.500 21.s = (9 + √2)/4 = 2.603+ 22.s = 8/3 = 2.666+ 23.s = 14/5 = 2.800 (MM) 24.s = (10 + √2)/4 = 2.853+ 25.s = 2.878+ (JD) 26.s = 2.934+ (JD) 36.s = 3 39.s = 10/3 = 3.333+ (JD) 40.s = 3.447+ (JD) 42.s = 7/2 = 3.500 (JD) 43.s = (13 + √2)/4 = 3.603+ (JD) 45.s = 11/3 = 3.666+ (JD) 46.s = 3.786+ (JD) 47.s = 3.824+ (JD)

S Tetromino
 4.s = 1 5.s = 19/13 = 1.461+ 7.s = 3/2 = 1.500 8.s = 4(4 + √3)/13 = 1.763+ 9.s = 1.931+ (MM) 10.s = (5 + 2√2)/4 = 1.957+ 16.s = 2 17.s = 1 + √2 = 2.414+ 18.s = 2.434+ (MM) 22.s = 5/2 = 2.500 23.s = 2.793+ (JD) 24.s = (7 + √2)/3 = 2.804+ 25.s = 2.853+ (JD) 26.s = 2.895+ (JD) 27.s = 2.948+ (JD) 36.s = 3 37.s = 3 + √2/4 = 3.353+ (JD) 38.s = 3.427+ (JD) 40.s = 3.472+ (JD) 45.s = 7/2 = 3.500 (JD) 47.s = 3.767+ (JD) 48.s = 3.871+ (JD) 49.s = 3.941+ (JD)

I Tetromino
 4.s = 1 5.s = 5/4 = 1.250 6.s = 3/2 = 1.500 7.s = 7/4 = 1.750 8.s = 1.906+ (MM) 9.s = (2 + 7√2)/6 = 1.983+ (MM) 16.s = 2 18.s = 9/4 = 2.250 19.s = (11 + √2)/5 = 2.482+ 20.s = 5/2 = 2.500 21.s = (12 + √2)/5 = 2.682+ 22.s = 11/4 = 2.750 23.s = 2.807+ 24.s = 2.862+ (JD) 25.s = 2.898+ (JD) 26.s = 2.930+ 27.s = 2.985+ (JD) 36.s = 3

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/16/15.