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?

ANSWERS

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.