# Problem of the Month (June 2010)

The problem of how many unit squares can be packed into a square of a certain size has been well-studied (see one of my packing pages or my paper in the Electronic Journal of Combinatorics). A natural related question is how many unit squares can be packed inside rectangles. We let s(x,y) denote the maximum number of non-overlapping unit squares that will fit inside an x × y rectangle. We also let Sn denote the portion of the first quadrant with x≥1, y≥1, and s(x,y)≤n. Then S1 = { x<2 and y<2 }. What is S2? What about Sn for larger n?

The same question can be asked about circles packed in rectangles (the appropriate packing page for circles in squares is here, though you might also want to look at these animations). Let c(x,y) denote the maximum number of non-overlapping circles with diameter 1 that will fit inside an x × y rectangle. Let Cn denote the portion of the first quadrant with x≥1, y≥1, and with c(x,y)≤n. Then C1 = { (x-1)2+(y-1)2<1 }. What is C2? What are Cn for larger n?

This month's solvers include Maurizio Morandi, Evert Stenlund, and David W. Cantrell.

Here are the results known for squares. The limiting packing shows how more squares can fit if you cross the boundary curve.

nBoundary Curves of SnLimiting PackingLargest Aspect RatioAuthor
1x = 21
2x = 3(1+2√2)/6 = .638+
x + 2y = 4 + 2√21/√2 = .707+
y = 21
3x = 4.470+
x = 2 [ 2 - 3 t2 - t3 + (t2 - 2 t + 2)√(1 - t2) ] / (2 - 3 t2)
y = 2 [ 1 - t2 - t3 + √(1 - t2)3 ] / (2 - 3 t2)
(1+2√2)/7 = .546+(MM)
y = 21
4x = 5(6√2-1)/20 = .374+(MM)
x + 4y = 4 + 6√2(2+3√2)/14 = .445+(MM)
y = 22/3 = .666+(MM)
x = 3(1+√2)/3 = .804+(MM)
x + y = 4 + √21(MM)
5x = 6.310+(MM)
x = 2 [ 3 - 2 t - 5 t2 - 2 t3 + 2(t2 - 2 t + 3)√(1 - t2) ] / (3 - 5 t2)
y = 2 [ 1 - t2 - 2 t3 + 2 √(1 - t2)3 ] / (3 - 5 t2)
(3+4√2)/23 = .376+(MM)
y = 22/3 = .666+(MM)
x = 31(MM)
6x = 7(10√2-3)/42 = .265+(MM)
x + 6y = 4 + 10√2(4+5√2)/34 = .325+(MM)
y = 21/2 = .500(MM)
x = 4 (1+√2)/4 = .603+(MM)
x + y = 5 + √2(6-3√2)/2 = .878+(MM)
y = 31(MM)
7x = 8.231+(MM)
x = 2 [ 4 - 6 t - 7 t2 - 3 t3 + 3(t2 - 2t + 4)√(1 - t2) ] / (4 - 7t2)
y = 2 [ 1 - t2 - 3t3 + 3√(1 - t2)3 ] / (4 - 7t2)
(5+6√2)/47 = .286+(MM)
y = 21/2 = .500(MM)
x = 41/√2 = .707+(MM)
x = [ 2t2 + 2t3 + (1 - 2t)√(1 - t2) ] / t3
y = (2t2 + 2t - 1) / t2
.795+(MM)
(x-2)2 + (y-2)2 = 43(2-√3) = .803+(MM)
y = 31(MM)
8x = 9(14√2-5)/72 = .205+(MM)
x + 8y = 4 + 14√2(6+7√2)/62 = .256+(MM)
y = 22/5 = .400(MM)
x = 5(1+√2)/5 = .482+(MM)
x + y = 6 + √2(2+6√2)/17 = .616+(MM)
x + 2y = 6 + 3√21/√2 = .707+(MM)
y = 31(MM)

Here are the results known for circles. The limiting packing shows how one more circle can fit if you cross the boundary curve.

nBoundary Curves of CnLimiting PackingLargest Aspect RatioAuthor
1(x-1)2 + (y-1)2 = 11
2(x-1)2 + 4(y-1)2 = 4(2+√3)/4 = .933+
(2x-2+√3)2 + (2y-2+√3)2 - 4√3 x y = (2-√3)21(MM)
3(x-1)2 + 9(y-1)2 = 9(2+√3)/5 = .746+
(x-2)2 + (y-1)2 = 11
4(x-1)2 + 16(y-1)2 = 16(2+√3)/6 = .622+
4(x-1+√3)2 + (4y-4+√3)2 - 8√3(xy-1) = 19√3-1 = .732+(MM)
(x-1)2 + (y-1)2 = 41
5(x-1)2 + 25(y-1)2 = 25(2+√3)/7 = .533+
(x-3)2 + (y-1)2 = 12/3 = .666+
(x-1)2 + 4(y-2)2 = 45(√3-1)/4 = .915+
4(x-1)2 + 9(y-1)2 = 361
6(x-1)2 + 36(y-1)2 = 36(2+√3)/8 = .466+
(x-2+√3)2 + (2y-2+√3)2 - 2√3 x y = (2-√3)24-2√3 = .535+(MM)
2x - √[y (4 - y)] - 2 √[(1 + y)(3 - y)] = 2(2+2√3)/(4+4√12) = .932+(MM)
4x - √3 y - 2 √[y (4 - y)] - √[(1 + y)(3 - y)] = 4 - √3.981+(MM)
(x + 2y)(1 + √3) - x √[(y - 1)(3 - y)] +
(1 - √3 - y + √[3 (y - 1)(3 - y)]) √[(x - 1)(3 - x)] - √3 x y = 3 + 2√3
1(MM)
7(x-1)2 + 49(y-1)2 = 49(2+√3)/9 = .414+(MM)
(x-4)2 + (y-1)2 = 11/2 = .500(MM)
(x-1)2 + 9(y-2)2 = 95/(2+3√3) = .694+(MM)
21(x-1+2/√3)2 + (7y-7+2√3)2 - 28√3(xy - 1) = 103(217+13√651)/(217+38√217) = .706+(DC)
(x-1)2 + 4(y-1)2 = 16(1+√3)/3 = .910+(MM)
(2x-2+√3)2 + (2y-2+√3)2 - 4√3 x y = 10 - 4√31(MM)
8(x-1)2 + 64(y-1)2 = 64(2+√3)/10 = .373+(MM)
4(x-3+√3)2 + (4y-4+3√3)2 - 8√3 x y = (3√3-4)21-1/√3 = .422+(MM)
x + 2y - (x - 1)√[(y - 1)(3 - y)] - (y - 1)√[(x - 1)(5 - x)] = 4(6+√3+3√((3+4√3)/13))/16 = .647+(MM)
(x - 1)√[(2y - √3)(4 + √3 - 2y)] + 2(y - 2)√[x (4 - x)] - x + 2(2 - √3)y = 5 - 2√3.647+(MM)
2x - 5 = 2√[4-(y-1)2] + (y-1)√[(3-(y-1)2)/(1+(y-1)2)].665+(DC)
4(x-1)2 + 25(y-1)2 = 1002(1+√3)/7 = .780+(MM)
4(x-3)2 + (y-1)2 = 41(MM)
9(x-1)2 + 81(y-1)2 = 81(2+√3)/11 = .339+(DC)
(x-5)2 + (y-1)2 = 12/5 = .400(DC)
(x-1)2 + 16(y-2)2 = 165/(2+4√3) = .560+(DC)
9(x-1)2 + 16(y-2)2 = 144.587+(DC)
(x-2)4 (8y2-20y+13) + 8(x-2)2 (y-1)2 (8y2-28y+17) + 16(y-1)4 (8y2-36y+37) = 0.590+(DC)
very complicated2(1+√3)/7 = .596+(DC)
16(x-1)4 + 8(x-1)2 (17y2-36y-48) + (15y2-28y-48)2 == 0(2+2√3)/(6+√(2√3)) = .695+(DC)
2(x-1) = √3(y-1) + √[(4-y)y] + √[(3-y)(1+y)].737+(DC)
very complicated(21+2√3)/33 = .741+(DC)
very complicated.754+(DC)
256(y-1)8 + 128(x2-8x+4) (y-1)6 + 16(7x4-76x3+278x2-344x+144) (y-1)4 - 8(x-1) (x-4) (3x4-15x3-4x2+68x-64) (y-1)2 + (x-1)2 (x-4)2 (3x2-15x+16)2 = 0(4+√(2√3))/(4+2√3) = .785+(DC)
very complicated.801+(DC)
very complicated.806+(DC)
x6 - 12x5 + 3x4 (y2-4y+22) - 8x3 (3y2-13y+28) + 3x2 (y4-8y3+44y2-128y+163) - 4x (3y4-26y3+96y2-184y+153) + y6 - 12y5 + 66y4 - 224y3 + 489y2 - 612y + 328 = 0.812+(DC)
16(x-1)2 (x2-2x-2) + 16(8x2-16x+5)y - 4(10x2-20x-21)y2 - 96y3 + 21y4 = 8(x-1) (4x2-8x-2+8y-3y2) √[y(4-y)]6/(2+√3) = .833+(DC)
16(x-1)2 + 9(y-1)2 = 144.939+(DC)
very complicated(37+7√7)/57 = .974+(DC)
very complicated1(DC)
10(x-1)2 + 100(y-1)2 = 100(2+√3)/12 = .311+(DC)
(x-4+√3)2 + 4(y-1+√3)2 2√3(xy-4) = 162/(4+√3) = .348+(DC)
(x-2)4 (8y2-20y+13) + 2(x-2)2 (y-1) (40y3-212y2+337y-173) + 64(x-2) (y-3) (y-1)3 + (y-1)2 (72y4-660y3+2173y2-3022y+1501) = 0.466+(DC)
(x-1)2 + 9(y-1)2 = 36(1+√3)/4 = .683+(DC)
(2x-4+√3)2 + 4(y-1+√3)2 4√3(xy-2) = 196-3√3 = .803+(DC)
very complicated.996+(DC)
very complicated1(DC)
11(x-1)2 + 121(y-1)2 = 121(2+√3)/13 = .287+(DC)
(x-6)2 + (y-1)2 = 11/3 = .333+(DC)
(x-1)2 + 25(y-2)2 = 255/(2+5√3) = .469+(DC)
9(x-1)2 + 25(y-1)2 = 225.488+(DC)
4(3-y) (y-1) (7-5x+x2-8y+2xy+3y2)2 = (3-8x-x2+14y+8xy+2x2y-19y2-4xy2+6 y3)2(3744+675√3+√(630291+1332640√3))/13292 = .498+(DC)
(x-4)4 (8y2-20y+13) + 2(x-4)2 (y-1)2 (8y2-28y+17) + (y-1)4 (8y2-36y+37) = 0.503+(DC)
2x - 5 = 4√[4-(y-1)^2] + (y-1)√[(3-(y-1)^2)/(1+(y-1)^2)].512+(DC)
4(x-1)2 + 49(y-1)2 = 1962(1+√3)/9 = .607+(DC)
4(x-4)2 + (y-1)2 = 43/4 = .750(DC)
(x-1)2 + 9(y-3)2 = 97/(2+3√3) = .972+(DC)
25(x-1)2 + 9(y-1)2 = 2251(DC)

Here is a picture of the small Cn:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/18/14.