1. Packing squares in squares
SASMS fall, 2008
David Rhee

2. 1. Introduction
Packing circles has been studied by Gauss and Kepler since 1600’s
Axel Thue(1890) and Thomas Hales (1998) proved that the hexagonal packing is the densest of all possible sphere packings in 2D and 3D
Packing squares began to be studied recently (~1970)

3. 1. Introduction
Let s(n) be the side length of smallest square into which we can pack n unit squares.
s(17) < 4.676
s(n) is non-decreasing.
s(n2)=n. .

4. 2. Upper bounds
Some upper bounds were first considered by Göbel in 1979
s(5)≤2+1/√2
s(10)≤3+1/√2
s(27)≤5+1/√2
s(84)≤9+1/√2

5. 2. Upper bounds
Other packings using squares tilted by 45° have been found since then.
s(10)≤3+1/√2
s(84)≤9+1/√2
s(28)≤4+1/√2
s(27)≤5+1/√2
s(52)≤7+1/√2

6. 2. Upper bounds
Some people improved the diagonal packings by slightly moving the squares
s(19)≤3+4√2/3
s(87)<9.8520

7. 2. Upper bounds
Generally, packings are more complicated. Many of these results were found by a new efficient algorithm found in 2005.
s(11)<3.8771
s(17)<4.6756
s(29)≤5.9344
s(68)≤15/2+√7/2
s(71)≤8.9633

8. 2. Upper bounds s(18)≤(7+√7)/2 Hämäläinen (1980) Gustafson (1981)
Cantrell (2002)
Gensane and Ryckelynck (2004)

9. 2. Upper bounds
It was conjectured that s(n2-n)=n. This was disproved by Lars Cleemann.
s(172-17)<17

10. 2. Upper bounds
W(s):=s2-max{n:s(n)≤s}, then W(s) is O(s7/11) (Erdös and Graham, 1975)
W(s)=O(s(3-√3)/2+ε) for every ε>0 (Montgomery, 1978?)
W(s) is not O(sα) when α<1/2 (Roth and Vaughan, 1978)

11. 3. Lower bounds
Lemma 1. (corner) Any unit square inside the first quadrant whose center is in [0,1]2 contains the point (1,1).
Lemma 2. (edge) Let 0<x≤1, 0<y≤1, and x+2y<2√2. Then any unit square inside the first quadrant whose center is contained in [1,1+x]x[0,y] contains either the point (1,y) or the point (1+x,y).

12. 3. Lower bounds
Lemma 3. (triangle) If the center of a unit square u is contained in ABC, and each side of the triangle has length no more than 1, then u contains A, B, or C.

13. 3. Lower bounds
Lemma 4. (rectangle) If the center of a unit square u is contained in the rectangle R=[0,1]x[0,0.4], then u contains a vertex of R.

14. 3. Lower bounds
Lemma 5. (trapezoid) If the unit square u has its center in [0,1]2, u is completely above the x-axis, and u does not contain (0,1) and (1,1) then u covers the following:
the segment from (0.2,1) to (0.8,1)
some point (0,y) for 1/2≤y≤1 and some point (1,y) for 1/2≤y≤1.
either the point (0,√2-1/2) or the point (1,√2-1/2).

15. 3. Lower bounds To show s(n)≥k:
Find a set of n-1 “unavoidable” points in a square with side length k
Shrink this diagram by a factor of (1-ε/k) to get a square with side length k-ε
Unit squares must contain one of these n-1 points in the interior, so we can pack at most n-1 squares
Therefore, s(n)>k-ε

16. 3. Lower bounds s(2)=s(3)=2 (1,1) is unavoidable in [0,2]2 by lemma 1
P={ (1,1), (1,1+1/√2), (1+1/√2,1), (1+1/√2,1+1/√2) } is unavoidable in [0,2+1/√2]2 by lemma 1, 2, 3

17. 3. Lower bounds We can show other lower bounds using the same method

18. 3. Lower bounds Some lower bound attained by this method are not sharp

19. 3. Lower bounds
s(7)=3
At least two squares have their centers in the regions containing question marks
There are 2 possible placements of these 2 squares

20. 3. Lower bounds s(n2+ n/2 +1) ≥ 2√2 + 2(n-2)/√5 (Green, 2000)
s(n2-2)=s(n2-1)=n (Nagamochi, 2005)
Conjecture: If s(n2-k)=n, then s((n+1)2-k)=n+1

21. 3.Lower Bounds

22. 4. More problems
Packings of many shapes other than squares are studied

23. 4. More problems Packing circles into general rectangle
Rigid packing of circles into largest square
Packing cubes into cubes

24. References
- retrived Nov. 8, 2008
- retrived Nov. 8, 2008