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/a² and n/b² are integers which add to c², 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 ≤ n², then S(n²-1,k) = 1 - (1-S(0,k))/n² unless S(n²-1,k) = 1. That is, the n²-1 squares should have length 1/n. But this is not true for many pairs.

David Cantrell improved several of my packings in 2007.

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) ≥ 44/45 = .977+	S(23,23) ≥ .982+ (David Cantrell)	S(24,24) ≥ 624/625 = .998+
S(25,25) = 1	S(26,26) ≥ 221/225 = .982+	S(27,27) ≥ 351/361 = .972+ (David Cantrell)
S(28,28) ≥ 1820/1849 = .984+ (David Cantrell)	S(29,29) ≥ 5945/6084 = .977+	S(30,30) ≥ 29/30 = .966+
S(31,31) ≥ 2263/2304 = .982+	S(32,32) = 1	S(33,33) ≥ 583/588 = .991+
S(34,34) ≥ 3485/3528 = .987+	S(35,35) ≥ 1295/1296 = .999+	S(36,36) = 1
S(37,37) ≥ 2257/2304 = .979+	S(38,38) ≥ 950/961 = .988+ (David Cantrell)	S(39,39) ≥ 195/196 = .994+
S(40,40) ≥ 80/81 = .987+	S(41,41) ≥ .976+ (David Cantrell)	S(42,42) ≥ .980+ (David Cantrell)
S(43,43) ≥ .979+ (David Cantrell)	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) ≥ .974+ (David Cantrell)	S(53,53) ≥ 3922/3969 = .988+	S(54,54) ≥ 783/800 = .978+
S(55,55) ≥ .980+ (David Cantrell)	S(56,56) ≥ 119/121 = .983+ (David Cantrell)	S(57,57) ≥ 95/96 = .989+
S(58,58) ≥ .987+ (David Cantrell)	S(59,59) ≥ 2006/2025 = .990+	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

This is a graph of the best lower bounds for S(n,n):

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