This month's problem is this: If you are pack n squares of one size and m squares of another (possibly equal) size inside a unit square, what is the largest area they can cover? What about using more than 2 sizes of squares?

Brendan Owen and all the solvers noted that if n, m, or n+m is square, then S(n,m)=1. Also, If there are positive integers a, b and c such that m/a^{2} and n/b^{2} are integers which add to c^{2}, then S(n,m)=1.

Joseph DeVincentis gave packings for n,m ≤ 11, which were mostly optimal.

Trevor Green gave packings for n ≤ 20 and m ≤ 5, which were mostly optimal.

Ulrich Schimke gave packings for n,m ≤ 5, which were mostly optimal.

Green and Schimke independently conjectured that if 0 ≤ k ≤ n^{2}, then S(n^{2}-1,k) = 1 - (1-S(0,k))/n^{2} unless S(n^{2}-1,k) = 1. That is, the n^{2}-1 squares should have length 1/n. But this is not true for many pairs.

David Cantrell improved several of my packings in 2007.

Maurizio Morandi improved several packings in 2012 and 2014.

Here are the best known lower bounds for S(n,n).

S(1,1) = 1 | S(2,2) = 1 | S(3,3) = 15/16 = .9375 | S(4,4) = 1 |

S(5,5) ≥ 125/144 = .868+ | S(6,6) ≥ .862+ | S(7,7) ≥ 35/36 = .972+ | S(8,8) = 1 |

S(9,9) = 1 | S(10,10) ≥ 65/72 = .902+ | S(11,11) ≥ 187/200 = .935 | S(12,12) ≥ 24/25 = .960 |

S(13,13) ≥ 377/392 = .961+ | S(14,14) ≥ 35/36 = .972+ | S(15,15) ≥ 255/256 = .996+ | S(16,16) = 1 |

S(17,17) ≥ 221/225 = .982+ | S(18,18) = 1 | S(19,19) ≥ 247/256 = .964+ (David Cantrell) | S(20,20) = 1 |

S(21,21) ≥ 609/625 = .974+ | S(22,22) ≥ 286/289 = .989+ (Maurizio Morandi) | S(23,23) ≥ .982+ (Maurizio Morandi) | S(24,24) ≥ 624/625 = .998+ |

S(25,25) = 1 | S(26,26) ≥ 221/225 = .982+ | S(27,27) ≥ 74/75 = .986+ (Maurizio Morandi) | S(28,28) ≥ 952/961 = .990+ (Maurizio Morandi) |

S(29,29) ≥ 5945/6084 = .977+ | S(30,30) ≥ 39/40 = .975 (Maurizio Morandi) | S(31,31) ≥ 775/784 = .988+ (Maurizio Morandi) | S(32,32) = 1 |

S(33,33) ≥ 583/588 = .991+ | S(34,34) ≥ .991+ (Maurizio Morandi) | S(35,35) ≥ 1295/1296 = .999+ | S(36,36) = 1 |

S(37,37) ≥ 481/484 = .993+ (Maurizio Morandi) | S(38,38) ≥ 950/961 = .988+ (David Cantrell) | S(39,39) ≥ 195/196 = .994+ | S(40,40) ≥ 80/81 = .987+ |

S(41,41) ≥ 3485/3528 = .987+ (Maurizio Morandi) | S(42,42) ≥ 119/120 = .991+ (Maurizio Morandi) | S(43,43) ≥ 1075/1089 = .987+ (Maurizio Morandi) | S(44,44) ≥ 440/441 = .997+ |

S(45,45) = 1 (David Cantrell) | S(46,46) ≥ 391/392 = .997+ | S(47,47) ≥ 1222/1225 = .997+ | S(48,48) ≥ 2400/2401 = .999+ |

S(49,49) = 1 | S(50,50) = 1 | S(51,51) ≥ 255/256 = .996+ | S(52,52) = 1 (Maurizio Morandi) |

S(53,53) ≥ 3233/3249 = .995+ (Maurizio Morandi) | S(54,54) ≥ 1836/1849 = .992+ (Maurizio Morandi) | S(55,55) ≥ 2035/2048 = .993+ (Maurizio Morandi) | S(56,56) ≥ 340/343 = .991+ (Maurizio Morandi) |

S(57,57) ≥ 471/475 = .991+ (Maurizio Morandi) | S(58,58) ≥ 2378/2401 = .990+ (Maurizio Morandi) | S(59,59) ≥ 20591/20736 = .993+ (Maurizio Morandi) | S(60,60) ≥ 255/256 = .996+ |

S(61,61) ≥ .992+ (David Cantrell) | S(62,62) ≥ 1147/1152 = .995+ | S(63,63) ≥ 4095/4096 = .999+ | S(64,64) = 1 |

S(65,65) ≥ 3601/3645 = .987+ (Maurizio Morandi) | S(66,66) ≥ 2860/2883 = .992+ (Maurizio Morandi) | S(67,67) ≥ 1675/1681 = .996+ (Maurizio Morandi) | S(68,68) ≥ 10132/10201 = .993+ (Maurizio Morandi) |

S(69,69) ≥ 299/300 = .996+ (Maurizio Morandi) | S(70,70) ≥ 6230/6241 = .998+ (Maurizio Morandi) | S(71,71) ≥ 4331/4356 = .994+ (Maurizio Morandi) | S(72,72) = 1 |

S(73,73) ≥ 25769/25921 = .994+ (Maurizio Morandi) | S(74,74) ≥ 10249/10368 = .988+ (Maurizio Morandi) | S(75,75) ≥ 3250/3267 = .994+ (Maurizio Morandi) | S(76,76) ≥ 323/324 = .996+ (Maurizio Morandi) |

S(77,77) ≥ 6545/6561 = .997+ (Maurizio Morandi) | S(78,78) ≥ 3445/3456 = .996+ (Maurizio Morandi) | S(79,79) ≥ .996+ (Maurizio Morandi) | S(80,80) = 1 |

S(81,81) = 1 | S(82,82) ≥ 2788/2809 = .992+ (Maurizio Morandi) | S(83,83) ≥ 8051/8100 = .993+ (Maurizio Morandi) | S(84,84) ≥ 1365/1369 = .997+ (Maurizio Morandi) |

S(85,85) ≥ 1445/1458 = .991+ (Maurizio Morandi) | S(86,86) ≥ 1247/1250 = .997+ (Maurizio Morandi) | S(87,87) ≥ 3161/3174 = .995+ (Maurizio Morandi) | S(88,88) ≥ 440/441 = .997+ (Maurizio Morandi) |

S(89,89) ≥ 48149/48400 = .994+ (Maurizio Morandi) | S(90,90) = 1 | S(91,91) ≥ 9919/10000 = .991+ | S(92,92) ≥ 391/392 = .997+ (Maurizio Morandi) |

S(93,93) ≥ 2697/2704 = .997+ (Maurizio Morandi) | S(94,94) ≥ 799/800 = .998+ | S(95,95) ≥ 323/324 = .996+ | S(96,96) ≥ 2400/2401 = .999+ (Maurizio Morandi) |

S(97,97) ≥ 8633/8649 = .998+ (Maurizio Morandi) | S(98,98) = 1 | S(99,99) ≥ 9999/10000 = .999+ | S(100,100) = 1 |

Here are the best known lower bounds for S(n,m). Click on the values for pictures.

n \ m | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | |||||||||||||||||

2 | 1 | 1 | ||||||||||||||||

3 | 1 | 7/8 | 15/16 | |||||||||||||||

4 | 1 | 1 | 1 | 1 | ||||||||||||||

5 | 1 | 13/16 | .920+ | 1 | 125/144 | |||||||||||||

6 | 1 | 8/9 | 1 | 1 | 74/81 | .862+ | ||||||||||||

7 | 1 | 1 | 17/18 | 1 | 11/12 | 17/18 | 35/36 | |||||||||||

8 | 1 | 1 | 35/36 | 1 | .964+ | 26/27 | 1 | 1 | ||||||||||

9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||||||||

10 | 1 | 29/32 | .931+ | 1 | 15/16 | 1 | 43/45 | .969+ | 1 | 65/72 | ||||||||

11 | 1 | .855+ | .932+ | 1 | 1 | 49/50 | 211/225 | .970+ | 1 | 139/144 | 187/200 | |||||||

12 | 1 | 7/8 | 24/25 | 1 | 19/20 | 1 | 17/18 | 120/121 | 1 | 23/24 | .932+ | 24/25 | ||||||

13 | 1 | 15/16 | 1 | 1 | 137/144 | 213/225 | 115/121 | 141/144 | 1 | 71/72 | 63/64 | 1 | 377/392 | |||||

14 | 1 | 1 | 31/32 | 1 | 61/64 | 31/32 | 63/64 | 1 | 1 | 39/40 | 1 | 53/54 | 109/112 | 35/36 | ||||

15 | 1 | 31/32 | 63/64 | 1 | .980+ | 47/48 | 71/72 | 143/144 | 1 | 1 | .983+ | 145/147 | 253/256 | 127/128 | 255/256 | |||

16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||

17 | 1 | 1 | 19/20 | 1 | 113/121 | 249/256 | 80/81 | 1 | 1 | .962+ | 61/64 | 311/324 | .957+ | 143/144 | .985+ | 1 | 221/225 | |

18 | 1 | 1 | .943+ | 1 | 117/121 | 24/25 | 1 | 1 | 1 | 162/169 | 31/32 | 35/36 | 27/28 | 1 | .985+ | 1 | 1 | 1 |

Maurizio Morandi

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/14/14.