THE ELECTRONIC
JOURNAL OF COMBINATORICS
(2009), DS#7.
Abstract
Let s(n) be the side of the smallest square into which we can pack n unit squares. We present a history of this problem, and give the best known upper and lower bounds for s(n) for n≤100, including the best known packings. We also give relatively simple proofs for the values of s(n) when n = 2, 3, 5, 8, 15, 24, and 35, and more complicated proofs for n=7 and 14. We also prove many other lower bounds for various s(n).
Table of Contents
The problem of packing equal circles in a square has been around for some 40 years and has seen much recent progress [2]. The problem of packing equal squares in a square is only recently becoming well known. Results were less plentiful, as the computeraided methods available for circles did not generalize for squares, until recently when an effective algorithm was found [20]. We give a few packings which improve upon those in the literature, illustrate a technique for obtaining lower bounds, and exhibit the best known packings for less than one hundred squares.
Let s(n) be the side of the smallest square into which we can pack n unit squares. It is clear that √n ≤ s(n) ≤ √n, the first inequality coming from area considerations, and the second coming from the facts that s(n) is nondecreasing and s(n^{2})=n. It is not hard to show that s(2)=s(3)=2. It is a little harder to show that s(5)=2+1/√2 [7].
The number of claims far outweighs the number of published results in this area. Göbel says that Schrijver claims that Bajmóoczy proved s(7)=s(8)=3 [7]. Walter Stromquist claimed to have proved s(6)=3 and s(10)=3+1/√2, and claimed to know how to prove s(14)=s(15)=4 and s(24)=5 [13]. Trevor Green sent me a proof for s(6)=3. None of these proofs were published. Said El Moumni evidently proved s(7)=s(8)=3 and s(15)=4 [12] but no one was aware of this until recently. Finally, in 2002, Kearney and Shiu published a proof of s(6)=3 [9].
In 2003, Stromquist proved s(10)=3+1/√2 [18]. In 2005, Nagamochi proved that s(n^{2}2)=s(n^{2}1)=n [19]. Also in 2005, Thierry Gensane and Philippe Ryckelynck published an inflation algorithm for finding good packings and found the first computer packing that might be optimal. [20]. There are many other good packings thought to be optimal, but as of yet no proofs. Here we prove the values of s(n) for square n and n=2, 3, 5, 7, 8, 14, 15, 24, and 35.
Previous results can be found in Section 2. Reccent packings appear in Section 3. In Section 4 we prove some technical lemmas that we use in Section 5 to prove the values of s(n) mentioned above. Lists of the best known upper and lower bounds for s(n) are given in the Appendix. Many of the results given are taken from unpublished letters and manuscripts, and private communications.
Göbel was the first to publish on the subject [7]. He found that a^{2}+a+3+(a1)√2 squares can be packed in a square of side a+1+1/√2 by placing a diagonal strip of squares at a 45^{o} angle. This gives the best known packings for all values of a except for a=3 and a=6 (see Figure 1).

By unrotating some rotated squares in the corner, we get some alternate packings for n=10, 67, and 84. (see Figure 2). David Cantell noticed in 2005 that alternative packings exist for n=27, 38, 52, and 84 using a minimum number of rotated squares (see Figure 2) [17].


Charles Cottingham, who improved some of Göbel's packings for n≤49, was the first to use diagonal strips of width 2 [6]. Soon after he produced a packing of 19 squares with a diagonal strip of width 2, Robert Wainwright improved Cottingham's packing slightly (see Figure 4) [4]. In 2002, David Cantrell found some alternative packings for 19 squares [14].
s(19)≤3+4√2/3 Figure 4. 
In 1980, Evert Stenlund improved many of Cottingham's packings, and provided packings for n≤100 [6]. His packing of 66 squares uses a diagonal strip of width 3 (see Figure 5). In this packing, the diagonal squares touch only the squares in the upper right and lower left corners. Adding an "L" to this packing gives the best known packing of 83 squares.
s(66)≤3+4√2 Figure 5. 
The best known packings for many values of n are more complicated. Many seem to require packing with squares at angles other than 0^{o} and 45^{o}. In 1979, Walter Trump improved Göbel's packing of 11 squares (see Figure 6). Many people have independently discovered this packing. The original discovery has been incorrectly attributed to Gustafson and Thule [11]. The middle squares are tilted about 40.182^{o}, and there is a small gap between these squares. This packing is also rigid.
s(11)<3.8771 Figure 6. 
In 1980, Hämäläinen improved on Göbel's packing of 18 squares (see Figure 7) [6]. In 1981, Mats Gustafson found an alternative optimal packing of 18 squares (see Figure 7). The middle squares in these packings are tilted by an angle of arcsin((√71)/4)≈24.295^{o}. In 2002, David Cantrell found another alternative packing (see Figure 7) [14] that is useful in building the best known packing for n=68 (see Figure 11). In 2004, the computer program of Gensane and Ryckelynck found yet another alternative packing (see Figure 7) [20].

In [3], Erdös and Graham define W(s)=s^{2}max{n:s(n)≤s}. Thus W(s) is the wasted area in the optimal packing of unit squares into an s × s square. They show (by constructing explicit packings) that W(s)=O(s^{7/11}). In [10], it is mentioned that Montgomery has improved this result to W(s)=O(s^{3√3/2+ ε} ) for every ε>0.
In [10], Roth and Vaughan establish a nontrivial lower bound for W(s). They show that if s(ss)>1/6, then W(s)≥ 10^{100}√(s  ss+1/2  ). This implies that W(s) is not O(s^{α}) when α<1/2.
It was conjectured that s(n^{2}n)=n whenever n is small. The smallest known counterexample of this conjecture, due to Lars Cleemann, is s(17^{2}17)<17. That is, 272 squares can be packed into a square of side 17 in such a way that the the square can be squeezed together slightly (see Figure 8). Three squares are tilted by an angle of 45^{o}, and the other tilted squares are tilted by an angle of arctan(8/15).
s(272)<17 Figure 8. 
We can generalize the packings in Figure 3 by placing the central square a little off center. We can pack 2a^{2}+2a+b^{2} squares in a rectangle with sides a+1/2+b/√2 and a+3/2+b/√2. Adding a column of squares to the side of this, we get a packing of 2a^{2}+4a+b^{2}+1 squares in a square of side a+3/2+b/√2. This gives the best known packings for 26 and 85 squares (see Figure 9).

Stenlund also modified a diagonal strip of width 4 to pack 87 squares. In 2002, David Cantrell changed the angles slightly to give a minutely better packing (see Figure 10) [14]. There is a thin space between two of the diagonal strips. Compare this with the packing of 19 squares in Figure 4.
s(87)<9.8520 Figure 10. 
In 2002, David Cantrell found packings of 53 and 68 squares that are slightly better than Göbel's (see Figure 11) [14].

In 1980, Pertti Hämäläinen improved Göbel's packing of 17 squares using a different arrangement of squares at a 45^{o} angle. But in 1998, John Bidwell, an undergraduate student at the University of Hawaii, improved this packing (see Figure 12) [1]. It is the smallest example where the best known packing contains squares at three different angles.
Also in 1998, I improved the best known packing of 29 squares using a modified diagonal strip of width 2. A few months later, Bidwell improved my packing slightly [1]. In 2005, David Cantrell used a completely different idea to find a better packing [17]. But in 2004, the computer program of Thierry Gensane and Philippe Ryckelynck found what now stands as the best known packing for 29 squares (see Figure 12) [20]. This packing contains squares at no less than 6 different angles.

In 1997, I generalized Stenlund's packing of 41 squares in Figure 4 to give packings of 70 and 88 squares using strips of width 2. In 2002, David Cantrell modified these packings to give the best known packings (see Figure 13) [14].

In 1997, I modified a diagonal strip of 2 squares to get packings of 54 and 69 squares based on Wainwright's packing of 19 squares in Figure 4. In 2002, David Cantrell used his alternative packings of 19 squares to improve these packings [14]. He further improved the packing of 54 squares in 2005 (see Figure 14) [17]. Maurizio Morandi noticed some 5 years later that adding an "L" to this packing improved Cantrell's packing of 69 squares, but then he further improved the packing (see Figure 14) [21].

In 2002, David Cantrell used a completely new idea to find the best known packing of 39 squares (see Figure 15) [14].
We can generalize the packings in Figure 7 to provide the best known packings of 86 squares (see Figure 15). The angle of the tilted squares is the same as in that Figure.

In 1997, I improved the packing for 37 squares using a modified diagonal strip of width 3. In 2002, David Cantrell improved this slightly (see Figure 16) [14]. The slanted squares are tilted at angles of approximately 42.086^{o} and 45^{o}. Adding an ``L'' to this packing of 37 squares gives the best known packing of 50 squares.
In 1979, Charles Cottingham found a packing of 41 squares that was only improved in 2005 by David Cantrell (see Figure 16). [17]
In 2002, David Cantrell found the first packing of 55 squares in a square of side less than 8 [14]. He then improved this packing in 2005. (see Figure 16) [17].
Also in 2005, David Cantrell found the first packing of 71 squares in a square of side less than 9. (see Figure 16) [17].
In 2009, Károly Hajba found the first known packing of 51 squares in a square of side less than 7+1/√2. (see Figure 16) [16].

Finally, we make the following conjecture:
Conjecture 1. If s(n^{2}k)=n, then s( (n+1)^{2}k)=n+1.
Figure 17. 
the critical points of f(θ) are
Figure 18. 
Figure 19. 
Figure 20. 
Figure 21. 
Figure 22. 
To show that s(n)≥k, we will modify a method used by Walter Stromquist [13]. We will find a set P of (n1) points in a square S of side k so that any unit square in S contains an element of P (possibly on its boundary). Shrinking these by a factor of (1ε/k) gives a set P' of (n1) points in a square S' of side (kε) so that any unit square in S' contains an element in P' in its interior. Therefore no more than (n1) nonoverlapping squares can be packed into a square of side (kε), and s(n)>kε. Since this is true for all ε>0, we must have s(n)≥k.
We call P a set of unavoidable points in S. We now prove lower bounds on s(n) by showing that certain sets of points are unavoidable.
Theorem 1. s(2)=s(3)=2.
Proof: Consider a unit square u in [0,2]^{2}. Since the center of u is either in [0,1]^{2} or [0,1]x[1,2] or [1,2]x[0,1] or [1,2]^{2}, Lemma 1 shows that u contains the point (1,1). That is, the set P={ (1,1) } is unavoidable in [0,2]^{2} (see Figure 23).
s(2)≥2 Figure 23. 
s(5)≥2+1/√2 Figure 24. 
s(8)≥3 Figure 25. 
is unavoidable in [0,4]^{2} by Lemmas 1, 2, 3, and 4 (see Figure 26).
s(15)≥4 Figure 26. 
is unavoidable in [0,5]^{2} by Lemmas 1, 2, and 3 (see Figure 27).
s(24)≥5 Figure 27. 
is unavoidable in [0,5]^{2} by Lemmas 1, 2, and 3 (see Figure 28).
s(35)≥6 Figure 28. 
Figure 29. 

Figure 31. 


The other lower bounds known are probably not sharp. For example, Trevor Green has shown [8]:
Theorem 9. s(n^{2}+1) ≥ 2√2  1 + (n(n1)^{2} + (n1)√(2n)) / (n^{2}+1).
Theorem 10. s(n^{2}+n/2+1) ≥ 2√2 + 2(n2)/√5.
And Walter Stromquist has shown [18]:
Theorem 11. s(11) ≥ 2 + 4/√5.
Unavoidable sets illustrating some of the lower bounds on s(n) are shown in Figure 34.

n  s(n)  Optimal?  Figure  Author 
1  1  
24  2  
5  2+1/√2≈2.7072  Figure 1  Göbel  
69  3  
10  3+1/√2≈3.7072  Figure 1  Göbel  
11  ≈3.8771  Figure 6  Trump  
1213  4  
1416  4  
17  ≈4.6756  Figure 12  Bidwell  
18  7/2+1/2√7≈4.8229  Figure 7  Hämäläinen  
19  3+4/3√2≈4.8857  Figure 4  Wainwright  
2022  5  
2325  5  
26  7/2+3/2√2≈5.6214  Figure 9  Friedman  
27  5+1/√2≈5.7072  Figure 1  Göbel  
28  3+2√2≈5.8285  Figure 3  Göbel  
29  ≈5.9344  Figure 12  Gensane/Ryckelynck  
3033  6  
3436  6  
37  ≈6.5987  Figure 16  Cantrell  
38  6+1/√2≈6.7072  Figure 1  Göbel  
39  ≈6.8189  Figure 15  Cantrell  
40  4+2√2≈6.8285  Figure 3  Göbel  
41  ≈6.9473  Figure 16  Cantrell  
4246  7  
4749  7  
50  ≈7.5987  
51  ≈7.7044  Figure 16  Hajba  
52  7+1/√2≈7.7072  Figure 1  Göbel  
53  ≈7.8231  Figure 11  Cantrell  
54  ≈7.8488  Figure 14  Cantrell  
55  ≈7.9871  Figure 16  Cantrell  
5661  8  
6264  8  
65  5+5/√2≈8.5356  Figure 3  Göbel  
66  3+4√2≈8.6569  Figure 5  Stenlund  
67  8+1/√2≈8.7072  Figure 1  Göbel  
68  15/2+√7/2≈8.8229  Figure 11  Cantrell  
69  ≈8.8287  Figure 14  Morandi  
70  ≈8.9121  Figure 13  Cantrell  
71  ≈8.9633  Figure 16  Cantrell  
7278  9  
7981  9  
82  6+5/√2≈9.5356  
83  4+4√2≈9.6569  
84  9+1/√2≈9.7072  Figure 1  Göbel  
85  11/2+3√2≈9.7427  Figure 9  Friedman  
86  17/2+√7/2≈9.8229  Figure 15  Friedman  
87  ≈9.8520  Figure 10  Cantrell  
88  ≈9.9018  Figure 13  Cantrell  
89  5+7/√2≈9.9498  Figure 3  Stenlund  
9097  10  
98100  10 
n  s(n)  Figure  Author 
23  2  Figure 23  Göbel 
5  2+1/√2≈2.7071  Figure 24  Göbel 
6  3  Kearney/Shiu  
7  3  Figure 29  Friedman 
8  3  Figure 25  Friedman 
10  3+1/√2≈3.7071  Stromquist  
1112  2+4/√5≈3.7888  Stromquist  
13  3.8437  Friedman  
14  4  Figure 31  Friedman 
15  4  Figure 26  Friedman 
1718  (40√2+19)/17≈4.4452  Figure 34  Green 
1920  6√24≈4.4852  Figure 34  Friedman 
21  4.7438  Friedman  
22  2√2+2≈4.8284  Green  
23  5  Nagamochi  
24  5  Figure 27  Friedman 
2627  2√2+(27+2√10)/13≈5.3918  Green  
2830  2√2+6/√5≈5.5117  Green  
31  5.6415  Green  
34  6  Nagamochi  
35  6  Figure 28  Friedman 
3739  2√2+(113+10√3)/37≈6.3506  Green  
4041  2√2+8/√5≈6.4061  Green  
4748  7  Nagamochi  
5053  2√2+(101+3√14)/25≈7.3174  Green  
6263  8  Nagamochi  
6568  2√2+71/13≈8.2899  Green  
7980  9  Nagamochi  
8285  2√2+(288+12√3)/41≈9.2667  Green 
[1] J. Bidwell, 1998, private communication.
[4] M. Gardner, "Mathematical Games", Scientific American (Oct 1979, Nov 1979, Mar 1980, Nov 1980).
[6] M. Gardner, 1998, private communication.
[8] T. Green, 2000, private communication.
[11] "Problem Ronden", Ronden (Apr 1980, Sep 1980, Dec 1980)
[13] W. Stromquist, "Packing Unit Squares Inside Squares III", unpublished manuscript, 1984.
[14] D. W. Cantrell, 2002, private communication.
[15] E. Friedman, "Erich's Packing Center", https://erichfriedman.github.io/packing/index.html.
[16] K. Hajba, 2009, private communication.
[17] D. W. Cantrell, 2005, private communication.
[18] W. Stromquist, Packing 10 or 11 Unit Squares in a Square, Elect. J. Comb. 10 #R8 (2003).
[19] H. Nagamochi, Packing Unit Squares in a Rectangle, Elect. J. Comb. 12 #R37 (2005).
[20] T. Gensane and P. Ryckelynck, Improved Dense Packings of Congruent Squares in a Square, Discrete Comput. Geom. 34 (2005) 97109.
[21] M. Morandi, 2010, private communication.