1. Square Chain Tilings
We say a tiling of a square by integer-sided squares is a square chain tiling if every square of side s>1 shares part of a side with a square of s–1. For each n, we are interested in finding the square chain tiling of a square of side n using the fewest number of squares.
What are the minimal square chain tilings for various squares? In particular, what is the fewest number of squares you can find for a chain tiling of a square of side 100?
What are the best chain tilings of rectangles by squares? How about equilateral triangle chain tilings?
2. Square Bridges
A square bridge is a collection of integer-sided squares on a grid so that a) every square but one rests partly on another square, and b) the tops of the squares which do not have a square resting on them are level. The width of a square bridge is the length of the top edge. For each n, we are interested in finding the square bridge of width n whose combined area is as small as possible.
What are the minimal square bridges for various n? Do squares of side 5 or greater ever appear in the optimal bridges? What is the smallest possible area of a square bridge of width 100?
3. Covering Squares Without Overlap
The September 2000 problem investigated covering the largest square using squares of sides 1 through n. This month we are interested in the largest square that can be covered by non-overlapping squares of sides 1 through n.
What are the best coverings for various n? In particular, what is the smallest n for which non-overlapping squares of sides 1 through n can cover a square of side 100? What is the largest area rectangle that non-overlapping squares of sides 1 through n can cover?
4. Sorted Square Packings
A sorted square packing is a packing of a square with integer-sided squares so that all squares above and to the right of a square are strictly smaller. For each n, we are interested in the sorted square packing of a square of side n with the smallest wasted area.
What are the best packings for various squares? In particular, what is the best sorted square packing of a square of side 100?
1![]() 1 square | 2![]() 4 squares | 3![]() 6 squares | 4![]() 7 squares | 5![]() 8 squares | 6![]() 11 squares | 7![]() 9 squares | 8![]() 15 squares |
9![]() 13 squares | 10![]() 15 squares | 11![]() 15 squares | 12![]() 15 squares (Berend van der Zwaag) | 13![]() 20 squares (Maurizio Morandi) |
14![]() 18 squares (George Sicherman) | 15![]() 17 squares (Jeremy Galvagni) | 16![]() 21 squares (George Sicherman) | 17![]() 23 squares |
18![]() 22 squares (Berend van der Zwaag) | 19![]() 27 squares (Berend van der Zwaag) | 20![]() 24 squares (Jeremy Galvagni) |
Jeremy Galvagni developed the idea of an efficiency ratio r(n) for a chain tiling of a square of side n: the area divided by the number of squares. He notes that r(kn) is an increasing function of k, since we can just tile a square of side kn with chain tilings of a square of side n. Presumably r(n) diverges to ∞, but how quickly?
Jeremy Galvagni also found a chain tiling of a square of side 100 containing 230 squares. Then Emilio Schiavi found a tiling using only 186 squares. Then in 2009, Berend van der Zwaag found a tiling using only 128 squares:
Finally, in 2011, Maurizio Morandi improved 2 parts of the tiling to reduce the number of squares:
Jeremy Galvagni also investigated triangle chain tilings. Here are his best results:
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|
Triangles | 1 | 4 | 6 | 7 | 11 | 14 | 16 | 12 | 19 | 18 | 26 |
2. Here are the best known small square bridges:
1![]() area 1 | 2![]() area 4 | 3![]() area 9 | 4![]() area 12 | 5![]() area 18 | 6![]() area 22 | 7![]() area 30 | 8![]() area 35 (Corey Plover) | 9![]() area 43 |
10![]() area 48 | 11![]() area 57 | 12![]() area 64 | 13![]() area 73 | 14![]() area 80 (Corey Plover) |
Corey Plover found this square bridge of width 100:
Corey Plover also sent me solutions for all n≤100 based on recursion. And he sent me an analysis of this problem when the squares do not have to lie on a grid. He used only unit squares to show that in this case, the area needed is no more than n+1+k2k where k=log2n
–1.
3. Here are the best known small non-overlapping square coverings:
1![]() side 1 | 2![]() side 2 | 3![]() side 3 | 4![]() side 4 | 5![]() side 6 (Maurizio Morandi) | 6![]() side 8.4744 (Maurizio Morandi) | 7![]() side 11 |
8![]() side 13 | 9![]() side 16 | 10![]() side 18 | 11![]() side 21 |
12![]() side 23 | 13![]() side 26.5583 (Maurizio Morandi) | 14![]() side 333/√122 = 30.1484 (Maurizio Morandi) |
15![]() side 34 (Vedran Glišić) | 16![]() side 37.1557 (Maurizio Morandi) | 17![]() side 41 (Maurizio Morandi) |
18![]() side 44 (Maurizio Morandi) | 19![]() side 48 (Maurizio Morandi) | 20![]() side 52 (Maurizio Morandi) |
21![]() side 55 (Maurizio Morandi) | 22![]() side 59 (Maurizio Morandi) |
4. Here are the best known small sorted square packings:
2![]() 2 wasted | 3![]() 3 wasted | 4![]() 5 wasted | 5![]() 5 wasted | 6![]() 8 wasted | 7![]() 5 wasted | 8![]() 9 wasted | 9![]() 8 wasted |
10![]() 8 wasted | 11![]() 11 wasted | 12![]() 13 wasted | 13![]() 12 wasted | 14![]() 13 wasted (VG) |
15![]() 16 wasted (VG) | 16![]() 14 wasted | 17![]() 17 wasted (MM) |
18![]() 21 wasted (MM) | 19![]() 19 wasted (MM) | 20![]() 17 wasted (MM) |
Corey Plover sent the following sorted square packing of a square of side 100.
In 2011, Vedran Glišić improved the packing:
Then a few weeks later, Maurizio Morandi did even better:
If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/12/11.