Find the minimum size square capable of bounding equal squares
arranged in any configuration. The first few cases are illustrated above (Friedman).
The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24,
and 35, in addition to the trivial cases of the square
numbers (Friedman).
If for some , it is conjectured that the
size of the minimum bounding square is for small . The smallest for which the conjecture is
known to be violated is
(with ).
The following table gives the smallest known side lengths for a square into which unit squares can be packed (Friedman
2005). An asterisk (*)indicates that a packing has been proven to be optimal.
exact
approx.
exact
approx.
1*
1
1
16*
4
4
2*
2
2
17
4.6755...
3*
2
2
18
4.822...
4*
2
2
19
4.885...
5*
2.707...
20
5
5
6*
3
3
21
5
5
7*
3
3
22
5
5
8*
3
3
23*
5
5
9*
3
3
24*
5
5
10*
3.707...
25*
5
5
11
3.877...
26
5.6214...
12
4
4
27
5.7072...
13
4
4
28
5.8285...
14*
4
4
29
5.9465...
15*
4
4
The best known packings of squares into a circle are illustrated above for the first few cases (Friedman).
The best known packings of squares into an equilateral triangle are illustrated above for the first few cases (Friedman).
The best packing of a square inside a pentagon,
illustrated above, is 1.0673....
equal squares
arranged in any configuration. The first few cases are illustrated above (Friedman).
The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24,
and 35, in addition to the trivial cases of the square
numbers (Friedman).
for some 
, it is conjectured that the
size of the minimum bounding square is 
for small 
. The smallest 
for which the conjecture is
known to be violated is 
(with 
).
unit squares can be packed (Friedman
2005). An asterisk (*)indicates that a packing has been proven to be optimal.
