The smallest values of n for which s(n) is not known are n = 26, 28, 32, 33, 34, 38, and 40. It is barely possible that s(26)=79, s(28)=88, and s(40)=149. These were thought to be impossible until the s(37)=133 packing was found. Can you find one of these packings? It is known that s(32)≤108, s(33)≤113, s(34)≤118, and s(38)≤139, and because the amount of wasted space is so small, these are believed to be optimal, but they might be 1 smaller. Can you prove these?

This month's problem is to consider the same packing problem for other shapes: Given a shape P find its packing sequence, in which the n^{th} term is the smallest magnification of P that contains copies of size 1 through n of P. For example, the domino seems to have packing sequence 1, 3, 4, 6, 8, 10, 12, 15, 18, 20, 23, .... Can you verify or extend this sequence? What about the packing sequences of other small polyominoes? What about equilateral triangles? What about other triangles?

I'll give $10 to the first person who finds a shape with s(9) = 17.

Bill Clagett gave the sequences for the 2x3, 2x5, and 3x5 rectangles, as well as correcting a few of my sequences.

Maurizio Morandi extended the sequence for the 2x5 rectangle.

Berend van der Zwaag sent solutions for the 3x4 rectangle. And he collected the $10 prize in 2011 for the packing below:

Here are the best known values of the packing sequences of various shapes, in lexigraphical order:

Shape | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 3 | 4 | 6 | 8 | 10 | 12 | 15 | 18 | 20 | 23 | 26 | 29 | 32 | 36 | 39 | 43 | |||||||||

1 | 3 | 4 | 6 | 8 | 10 | 12 | 15 | 18 | 20 | 23 | 26 | ||||||||||||||

1 | 3 | 5 | 6 | 8 | 10 | 13 | 15 | 18 | 21 | 24 | 26 | ||||||||||||||

1 | 3 | 5 | 6 | 8 | 10 | 13 | 15 | ||||||||||||||||||

1 | 3 | 5 | 6 | 8 | 11 | 13 | 15 | ||||||||||||||||||

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 43 | 47 | 50 | 54 | 58 | 62 | 66 | 71 | 75 | |

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 43 | 47 | 50 | 54 | 58 | 62 | 66 | 71 | 75 | |

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 43 | 47 | 50 | 54 | 58 | 62 | 66 | 71 | 75 | |

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 43 | 47 | 50 | 54 | 58 | 62 | 66 | 71 | 75 | |

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | |||||||||||||||||

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | 24 | 27 | ||||||||||||||

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 18 | 21 | ||||||||||||||||

1 | 3 | 5 | 7 | 9 | 11 | 13 | 16 | 18 | |||||||||||||||||

1 | 3 | 5 | 7 | 9 | 12 | 14 | 16 | 19 | |||||||||||||||||

1 | 3 | 5 | 7 | 9 | 12 | 14 | |||||||||||||||||||

1 | 3 | 5 | 7 | 9 | 12 | 14 | |||||||||||||||||||

1 | 3 | 5 | 7 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | ||||||||||||

1 | 3 | 5 | 8 | 10 | 12 | 14 | 16 | ||||||||||||||||||

1 | 3 | 6 | 8 | 10 | 12 | 15 | 18 | 21 | 24 | 27 | |||||||||||||||

1 | 3 | 6 | 8 | 10 | 12 | 15 | 18 | 21 | 24 | 27 | |||||||||||||||

1 | 3 | 6 | 8 | 10 | 12 | 15 | 18 | ||||||||||||||||||

1 | 4 | 6 | 8 | 10 | 12 | 14 | 17 | 19 | 22 | ||||||||||||||||

1 | 4 | 6 | 8 | 10 | 12 | 15 | |||||||||||||||||||

1 | 4 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 |

Here are the best known packings for squares:

And here are the best known packings for dominoes that are better:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/27/11.