Problem of the Month
(March 2014)

What is the smallest square that n squares of area 1 and m squares of area 2 can be packed into? Can you improve or extend these packings?


ANSWERS

Solutions were received from Maurizio Morandi, Jeremy Galvagni, and Joe DeVincentis.

m \ n012345678910
0 0 1
2 2 2
2+√2/2
3 3 3 3
3+√2/2
1 √2
1+√2 1+√2 1+√2
3 3
(5+√3)/2(MM)
2+√2 2+√2 1+2√2(MM)
4
2 2√2 2√2 2√2 3
2+√2 2+√2 2+√2 2+√2
3.883 (MM) 4 4
3 2√2 2√2 2+√2 2+√2 2+√2
8√2/3 (MM)
1+2√2 3.883 (MM)
4 4.297 (MM)
3+√2
4 2√2
5√2/2
1+2√2 1+2√2 1+2√2 1+2√2
4 4
4.346 (MM)
3+√2 3+√2
5 1+2√2
2+4√2/3(MM)
3.887 3.887 3.887
3√2 3+√2 3+√2 3+√2 3+√2
2+2√2
6 3√2 3√2 3√2 3√2 3√2
3+√2
4.711 (MM)
2+2√2 2+2√2 2+2√2 2+2√2
7 3√2 3√2 3√2 3+√2
1/2+3√2(MM)
2+2√2 2+2√2 2+2√2 2+2√2
5.214 (JD)
1+3√2
8 3√2 3√2 2+2√2 2+2√2 2+2√2
3+3/√2(MM)
5.215 5.215 5.215 (JD)
1+3√2 1+3√2
9 3√2
7√2/2
5.119 (MM)
1+3√2 1+3√2 1+3√2 1+3√2 1+3√2 1+3√2 1+3√2
4√2
10 1+3√2 1+3√2 1+3√2 1+3√2 1+3√2
2+7√2/3(MM)
5.302 (MM)
5.607 (JD)
5.613 (JD)
5.711 (MM)
3+2√2
11 5.482 5.482 (MM)
2+5/√2(MM) 2+5/√2(MM)
4/3+3√2(JD)
5.607 (JD)
4√2 4√2(MM)
5.760 (MM)
3+2√2 3+2√2
12 4√2 4√2 4√2 4√2 4√2 4√2 3+2√2 3+2√2 3+2√2 3+2√2
2+3√2
13 4√2 4√2 4√2 4√2 4√2
3+2√2
6.125 (MM)
2+3√2 2+3√2 2+3√2 2+3√2
14 4√2 4√2 4√2 3+2√2
1/2+4√2(MM)
2+3√2 2+3√2 2+3√2 2+3√2
6.521 (MM)
6.630
15 4√2 4√2 2+3√2 2+3√2 2+3√2
9/√2
6.521 (MM)
6.630 6.630 (MM)
1+4√2 1+4√2
16 4√2
9/√2
(3+7√2)/2(MM)
6.607 (MM)
(10+7√2)/3(MM)
1+4√2 1+4√2 1+4√2 1+4√2 1+4√2 1+4√2
17 6.611
(10+7√2)/3(MM)
1+4√2 1+4√2 1+4√2 1+4√2 1+4√2 1+4√2 1+4√2
6.827 6.827 (JD)
18 (7+√7)/√2 (7+√7)/√2 (7+√7)/√2 (7+√7)/√2 (7+√7)/√2 (7+√7)/√2
6.821 (JD)
6.886 (JD)
5√2 5√2 5√2
19 3√2+8/3
6.921 (JD)
6.922 (JD)
4/3+4√2(JD)
7.020 (JD)
7.021 (JD)
5√2 5√2 5√2 7.125 (MM)
3+3√2
20 5√2 5√2 5√2 5√2 5√2 5√2 5√2 5√2
7.174 (MM)
3+3√2 3+3√2

m \ n11121314151617181920
0 3.877
4 4 4 4 4
4.675
(7+√7)/2
3+4√2/3
5
1 4 4
4.362 (MM)
3+√2 3+√2 4+1/√2(MM)
4.839 (MM)
5 5 5
2 4.365 (MM)
3+√2 3+√2 3+√2 2+2√2 4+2/√5(MM)
5 5 5
5.350 (MM)
3 3+√2 3+√2 3+√2(MM)
2+2√2(MM)
5 5 5 5.297 (MM)
4+√2 4+√2
4 4.764 (MM)
2+2√2 7/2+√2(MM)
5 5 5
5.346 (MM)
4+√2 4+√2 4+√2(MM)
5 2+2√2
5 5 (MM)
1+3√2 5.378 (MM)
4+√2 4+√2 4+√2(MM)
5.764 (MM)
3+2√2
6 5 (MM)
1+3√2 1+3√2
4+√2 4+√2 4+√2(MM)
4√2 3+2√2 3+2√2 9/2+√2(MM)
7 1+3√2
4+√2 4+√2 4√2 4√2 4√2
3+2√2 3+2√2 6 6.214 (JD)
8 4+√2 4+√2(MM)
5.776 (MM)
3+2√2 3+2√2 3+2√2 3+2√2(MM)
6
6.215 (MM)
2+3√2
9 5.779 (MM)
3+2√2 3+2√2 3+2√2 6.085 (JD)
2+3√2 2+3√2 2+3√2 2+3√2 2+3√2
10 3+2√2 3+2√2 3+2√2(MM)
2+3√2 2+3√2 2+3√2 6.302 (MM)
5+√2 5+√2(MM)
1+4√2
11 6.179 (MM)
2+3√2 2+3√2 2+3√2 2+3√2 5+√2
6.607 (JD)
1+4√2 1+4√2(MM)
4+2√2
12 2+3√2 2+3√2 2+3√2 2+3√2(MM)
6.525 (JD)
1+4√2 1+4√2 6.764 (JD)
4+2√2 4+2√2
13 2+3√2 2+3√2(MM)
6.630 (JD)
1+4√2 1+4√2 1+4√2(MM)
4+2√2 4+2√2
5√2 5√2
14 6.630 (JD)
1+4√2 1+4√2 1+4√2(MM)
4+2√2 4+2√2(MM)
5√2 5√2 5√2
3+3√2
15 1+4√2 1+4√2 4+2√2 4+2√2 5√2 5√2 5√2 3+3√2 3+3√2 3+3√2
16 1+4√2
4+2√2 4+2√2(MM)
5√2 5√2 5√2(MM)
3+3√2 3+3√2 3+3√2 7.593 (MM)
17 5√2 5√2
7.190 (MM)
3+3√2 3+3√2 3+3√2 3+3√2 3+3√2(MM)
7.630 (JD)
2+4√2
18 7.193 (MM)
3+3√2 3+3√2 3+3√2 7.499 (JD)
2+4√2 2+4√2 2+4√2 2+4√2 2+4√2(MM)
19 3+3√2 3+3√2 3+3√2(MM)
2+4√2 2+4√2 2+4√2 2+4√2 2+4√2 5+2√2 5+2√2(MM)
20 7.593 (JD)
2+4√2 2+4√2 2+4√2 2+4√2 2+4√2 2+4√2(MM)
5+2√2(MM)
8.045 (JD)
1+5√2

Jeremy Galvagni noticed that multiplying the (m,n) packing by √2 gives a (n,4m) packing.


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/3/14.