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## Perfect Rectangles

Problem: Given four natural numbers mR<=mC,  tR<= tC,  find the smallest rectangle with integer sides (with ratio of mR:mC) which can be completely tiled with distinct rectangles whose sides have a ratio of tR:tC.

Here are some of the solutions I found.

The number inside a rectangles is the length of the smallest side. The dimension of the smallest ones can be deducted from the "signature" above each picture.
The "signature" represents very succintly the solution: if you imagine to start filling the large rectangle from the top left corner and to insert a tile in the leftmost place available, the signature represent the order in which you'll insert the tiles of a given size  (real size is given by (n*tR,n*tC)). The symbol "-" (minus) following the size means a horizontal tile, while "|" means a vertical tile.

### Summary

1:2 Tiles1:1, 1:2 1:3 1:41:52:3 3:4, 2:53:5 4:5
1:3 Tiles1:1, 1:21:32:3, 3:5
1:4 Tiles1:4
1:5 Tiles1:5
2:3 Tiles1:1, 2:3, 4:5
3:4 Tiles3:4

1:2 Tiles

1:1 tiled with 1:2

34x34 = (10) 13- 10- 11- 3- 7- 6- 5- 8| 1- 2|

48x48 = (10) 20- 17- 11- 6- 5- 1- 2| 14| 4- 8|

Plus all those obtained taking a  [1:2 tiled with 1:2] solution below of sides p and 2p and putting another rectangle of the same size on the top, obtaining a (2p)x(2p) square.

1:2 tiled with  1:2

18x36 = (8) 10- 8- 2- 3| 1| 4- 7| 9|

31x62 = (10)  17- 14- 3- 11- 10| 4| 6| 7- 1- 12|

Other solutions:
33x66 = (11) 17- 16- 8| 11- 6- 3- 13- 5- 4- 2| 10|
35x70 = (9) 19- 16- 8| 11- 4| 1- 15- 9- 10|
40x80 = (13) 14- 13| 11- 15- 7| 5- 4- 8| 1- 17| 6- 3| 20|
42x84 = (12) 25- 17- 7- 10- 1- 6- 13| 2- 4| 5| 3| 21|
42x84 = (12) 22- 20- 10| 16- 6- 3- 17- 9- 5- 4- 8| 2|
45x90 = (13) 20- 5- 10| 11- 14- 8- 3- 17- 16- 4- 12- 9- 18|
48x96 = (13) 30- 18- 10- 8- 2- 3| 1| 4- 7| 6- 12| 9| 24|
48x96 = (13) 25- 23- 3- 7- 13- 11- 5- 4| 1- 2| 6| 12| 24|
49x98 = (13) 18- 15- 16- 3- 6| 8| 25- 4| 24- 12| 1- 5- 10|
50x100 = (12) 26- 24- 4| 16- 2- 5| 11| 1| 6| 17| 10- 20|
50x100 = (14) 28- 22- 11| 9| 4- 12- 8| 2| 16| 1| 6- 3| 10- 20|
51x102 = (13) 26- 25- 1- 12| 20- 7- 4| 16- 15- 10| 8| 3- 6|
etc.etc.

1:3 tiled with 1:2

36x108 = (14) 20- 16- 8| 14- 12- 10- 2- 4| 11| 7| 6- 15| 3| 18|

44x132 = (17) 23- 21- 4| 13- 20- 11- 8- 16- 1- 7- 12- 9- 3- 2| 10| 6| 22|

Plus those, like the following two, which derive immediately from [1:1 with 1:2] solutions (plus a large square)
34x102 = (11) 34- 13- 10- 11- 3- 7- 6- 5- 8| 1- 2|
36x108 = (10) 36- 18- 10- 8- 2- 3| 1| 4- 7| 9|

1:4 tiled with 1:2

Solutions can be obtained putting together a solution of [1:2 with 1:2] (see above) with a 1:2 rectangle of the same size, like in
18x72 = (9) 18- 10- 8- 2- 3| 1| 4- 7| 9|

The smallest solution not of this kind is
35x140 = (14) 20- 15- 4- 11- 2| 12| 19- 16- 8| 18- 1- 17- 7- 14|

1:5 tiled with 1:2

Solutions can be obtained putting together a solution of [1:3 with 1:2] (see above) with a 1:2 rectangle, like in
36x180 = (15) 36- 20- 16- 8| 14- 12- 10- 2- 4| 11| 7| 6- 15| 3| 18|

No other (primitive) solutions  for R up to 41x205.

2:3 tiled with 1:2

18x27 = (7) 10- 8- 2- 3| 1| 4- 7|

34x51 = (11) 13- 10- 11- 3- 7- 6- 5- 8| 1- 2| 17|

3:4 tiled with 1:2

36x48 = (8) 19- 17- 2- 3- 6| 10| 1- 8|    and  36x48 = (8) 15- 11- 10- 1- 2| 5- 8| 18|

2:5 tiled with 1:2    (trivial!)

3:5 tiled with 1:2

48x80 = (11) 28- 20- 9- 11- 12- 3- 6| 1- 2- 4| 18|
48x80 = (12) 26- 22- 11| 14- 6- 3| 4- 12| 1| 2- 8- 13|
48x80 = (12) 26- 22- 11| 13| 7| 8- 3| 4| 15| 1| 5| 9|
48x80 = (12) 21- 12- 15- 9- 3- 5- 13- 1- 2| 7- 14| 24|

4:5 tiled with 1:2

32x40 = (10) 12- 4- 8| 11- 9- 6| 2- 7- 5- 10|

1:3 Tiles

1:1 tiled with 1:3

The following solution is obtained joining a 1:3 tile with a solution of [2:3 with 1:3] below. I'm not sure if this is also the minimal (w.r.t. side) one.

96x96 = (12) 32- 27- 18- 19- 9- 8- 1- 7- 13- 3| 6- 15|

1:2 tiled with 1:3

72x144 = (16) 30- 14| 25- 17- 8- 9- 11| 10| 3| 2- 7- 20| 1| 5- 4| 24|

1:3 tiled with 1:3

57x171 = (16) 30- 27- 9| 18- 12- 10- 17- 3- 7- 6- 5- 1- 4- 11| 8| 19|

64x192 = (11) 37- 27- 13- 14- 10- 9| 7- 24- 3| 23- 17-

Other solutions:
59x177 = (18) 20- 13| 15- 24- 2| 9- 26- 11| 14- 19- 4- 12| 10| 1- 3| 7| 8- 17|
63x189 = (14) 38- 25- 11- 14- 8- 3- 18- 2- 6| 17- 16- 4- 12- 21|
68x204 = (13) 39- 29- 12- 17- 19- 2- 6| 27- 5- 22- 10- 9- 3|
69x207 = (16) 44- 25- 11- 14- 8- 3- 5- 12- 19| 13| 6| 4| 7| 9- 20| 23|

2:3 tiled with 1:3

64x96 = (11) 27- 18- 19- 9- 8- 1- 7- 13- 3| 6- 15|

3:5 tiled with 1:3

63x105 = (13) 23- 4- 12| 19- 11- 10- 1- 3| 8- 13- 5- 15| 21|

1:4 Tiles

1:4 tiled with 1:4

40x160 = (12) 23- 17- 7- 10- 16- 6- 1- 5- 3- 13- 11- 4|

1:5 Tiles

1:5 tiled with 1:5

48x240 = (11) 28- 20- 8- 12- 19- 13- 4- 16- 6- 7- 5|

2:3 Tiles

1:1 tiled with 2:3

120x120 = (10) 28- 13- 19- 4- 6| 11- 2- 21- 12- 18|

2:3 tiled with 2:3

130x195 = (16) 29- 14- 13- 9- 6| 1- 21- 15- 26| 5- 8- 12| 3- 23- 2| 28|

4:5 tiled with 2:3

96x120 = (11) 28- 20- 8- 5- 7- 3- 2- 9- 12- 18| 6|

3:4 Tiles

3:4 tiled with 3:4

168x224 = (12) 29- 27- 13- 14- 18- 11- 7- 16- 1- 15- 9- 12|