# Problem of the Month (January 2006)

In August 2002, we searched for Partridge Tilings of squares and other shapes. This month we generalize and look for Sub-Partridge Tilings. We say there is a {a1, a2, ... an} tiling if we can tile a rectangle with a1 squares of side a1, a2 squares of side a2, . . . and an squares of side an. For example, a {4,6,8,10} tiling is shown below. What sub-partridge tilings exist?

We can also look for Border Sub-Partridge Tilings, a tiling of a rectangle with squares with a border of width 1 around each square. For example, a {1,3,4,5,7} border tiling is shown below. What border sub-partridge tilings exist?

Mike Reid proved three infinite families of sub-partridge tilings. The first, is {4,8,12,...4n}, as shown by the following tiling:

The second is {2,4,6,...2n}, as shown by the following tiling:

The third is {4,8,24,60,...4FnFn+1} (where Fn is the nth Fibonacci number), as shown by the following tiling:

Jeremy Galvagni showed that {a2bc,ab2c} tilings exist. He conjectured that these are the only sub-partridge tilings using only two sizes of squares.

Luke Pebody showed that this conjecture isn't true by finding the {35,70} tiling below. He and Emilio Schiavi showed that an {n,2n} tiling does not exist precisely when n is an odd prime or n=1, 9, 15, 21, 25, 27, or 33.

Patrick Hamlyn found many sub-partridge tilings with his tiling program.

Patrick Hamlyn and Jeremy Galvagni noted that border tilings are equivalent to tiling n-1 copies of each square of side n without borders.

Claudio Baiocchi notes that if there is a {a1, a2, . . . an} tiling, then there is a {k a1, k a2, . . . k an} tiling for every k, and that for every {a1, a2, . . . an}, there exists some k so that there is a {k a1, k a2, . . . k an} tiling.

Known Sub-Partridge Tilings
Largest
Square
Number
of Sizes
Tilings
42{2,4}
63{2,4,6}
82{4,8}
3{4,6,8}
4{2,4,6,8}
7{1,2,4,5,6,7,8}, {2,3,4,5,6,7,8}
8{1,2,3,4,5,6,7,8}
92{3,9}
6{4,5,6,7,8,9}
7{2,3,5,6,7,8,9}, {3,4,5,6,7,8,9}
8{1,2,3,4,6,7,8,9}, {1,2,3,5,6,7,8,9}, {1,2,4,5,6,7,8,9}, {2,3,4,5,6,7,8,9}
9{1,2,3,4,5,6,7,8,9}
104{4,6,8,10}
5{2,4,6,8,10}
10{1,2,3,4,5,6,7,8,9,10}
118{1,2,4,5,6,7,8,11}
11{1,2,3,4,5,6,7,8,9,10,11}
122{6,12}
3{4,8,12}
4{3,6,9,12}, {4,6,8,12}, {6,8,10,12}
5{4,6,8,10,12}
11{1,2,3,4,5,6,7,8,9,10,12}
12{1,2,3,4,5,6,7,8,9,10,11,12}
1313{1,2,3,4,5,6,7,8,9,10,11,12,13}
144{6,8,12,14}
14{1,2,3,4,5,6,7,8,9,10,11,12,13,14}
153{5,10,15}, {9,12,15}
4{6,9,12,15}
15{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
162{4,16}, {8,16}
3{4,8,16}, {8,12,16}
4{2,4,6,16}, {4,8,12,16}
8{2,4,6,8,10,12,14,16}
16{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}
1717{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}
182{6,18}, {12,18}
3{2,4,18}, {4,8,18}, {6,12,18}
9{2,4,6,8,10,12,14,16,18}
18{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
1919{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}
202{10,20}
3{10,15,20}, {12,16,20}
4{8,12,16,20}
5{4,8,12,16,20}
10{2,4,6,8,10,12,14,16,18,20}
20 {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
Patrick Hamlyn

Known Border Sub-Partridge Tilings
Largest
Square
Number
of Sizes
Tilings
52{2,5}
75{1,3,4,5,7}
6{2,3,4,5,6,7}
87{2,3,4,5,6,7,8}
8{1,2,3,4,5,6,7,8}
92{4,9}, {5,9}
3{5,8,9}
6{3,4,5,6,8,9}
7{1,2,4,6,7,8,9}
8{1,2,3,4,5,7,8,9}, {1,2,4,5,6,7,8,9}
9{1,2,3,4,5,6,7,8,9}
108{1,2,3,5,6,7,9,10}
112{3,11}
4{2,5,8,11}
7{1,3,4,5,7,9,11}
8{1,2,3,4,5,6,9,11}
132{6,13}
142{5,14}, {9,14}
3{5,9,14}
4{2,8,11,14}
154{5,8,9,15}
163{3,11,16}
172{8,17}
4{5,8,9,17}, {8,9,14,17}
183{8,17,18}
192{4,19}
3{4,9,19}
Patrick Hamlyn

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/31/06.