Find the minimum size square capable of bounding 
equal squares
arranged in any configuration. The first few cases are illustrated above (Friedman).
Among the nontrivial cases, packings which have been proven optimal include 2, 3,
5, 6, 7, 8, 10, 13, 14, 15, 22, 23, 24, 33, 34, and 35, in addition to the trivial
cases of the square numbers (Friedman and Ellsworth).
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
and Ellsworth). 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 | 5.8244... | |
| 14* | 4 | 4 | 29 | 5.9338... | |
| 15* | 4 | 4 |
The value listed for 
is the polynomial root 
(Ellsworth).
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....
