pentagonal tiling

[Tom Sirgedas]

Above is a tiling for 15 pinwheels (a pinwheel is 4 tiles in a pinwheel shape) tiled with 7 colors in a 8:8:8:9:9:9:9 ratio. So, the least used color makes up 0.133333 of the tiles.

I found solutions for up to 100 pinwheels with glucose (a SAT solver). I used MaxHS (a MAX-SAT solver) to find the optimal solution for finding a coloring with the smallest least-used-color-ratio.

maxhs:  (should be optimal for minimizing one color)

7  pinwheels = 0.142857 4:4:4:4:4:4:4
10 pinwheels = 0.125 5:5:6:6:6:6:6
12 pinwheels = 0.125 6:7:7:7:7:7:7
14 pinwheels = 0.142857 8:8:8:8:8:8:8
15 pinwheels = 0.133333 8:8:8:9:9:9:9
17 pinwheels = 0.132353 9:9:10:10:10:10:10
19 pinwheels = 0.131579 10:11:11:11:11:11:11
20 pinwheels = 0.125 10:11:11:12:12:12:12
22 pinwheels = 0.125 11:12:13:13:13:13:13
24 pinwheels = 0.125 12:14:14:14:14:14:14
25 pinwheels = 0.13 13:14:14:14:15:15:15
26 pinwheels = 0.134615 14:15:15:15:15:15:15
27 pinwheels = 0.12963 14:15:15:16:16:16:16
29 pinwheels = 0.12931 15:16:17:17:17:17:17
30 pinwheels = 0.125 15:17:17:17:18:18:18
31 pinwheels = 0.129032 16:18:18:18:18:18:18
32 pinwheels = 0.125 16:18:18:19:19:19:19

glucose: (arbitrary solution)

7  pinwheels = 0.142857 4:4:4:4:4:4:4
10 pinwheels = 0.125 5:5:6:6:6:6:6
12 pinwheels = 0.125 6:7:7:7:7:7:7
14 pinwheels = 0.142857 8:8:8:8:8:8:8
15 pinwheels = 0.133333 8:8:8:9:9:9:9
17 pinwheels = 0.132353 9:9:10:10:10:10:10
19 pinwheels = 0.131579 10:11:11:11:11:11:11
20 pinwheels = 0.1375 11:11:11:11:12:12:12
21 pinwheels = 0.142857 12:12:12:12:12:12:12
22 pinwheels = 0.125 11:12:13:13:13:13:13
24 pinwheels = 0.135417 13:13:14:14:14:14:14
25 pinwheels = 0.14 14:14:14:14:14:15:15
26 pinwheels = 0.134615 14:15:15:15:15:15:15
27 pinwheels = 0.12963 14:15:15:16:16:16:16
28 pinwheels = 0.142857 16:16:16:16:16:16:16
29 pinwheels = 0.137931 16:16:16:17:17:17:17
30 pinwheels = 0.141667 17:17:17:17:17:17:18
31 pinwheels = 0.137097 17:17:18:18:18:18:18
32 pinwheels = 0.132813 17:17:18:19:19:19:19
33 pinwheels = 0.136364 18:19:19:19:19:19:19
34 pinwheels = 0.132353 18:18:20:20:20:20:20
35 pinwheels = 0.142857 20:20:20:20:20:20:20
36 pinwheels = 0.138889 20:20:20:21:21:21:21
37 pinwheels = 0.141892 21:21:21:21:21:21:22
38 pinwheels = 0.138158 21:21:22:22:22:22:22
39 pinwheels = 0.141026 22:22:22:22:22:23:23
40 pinwheels = 0.1375 22:23:23:23:23:23:23
41 pinwheels = 0.140244 23:23:23:23:24:24:24
42 pinwheels = 0.136905 23:23:24:24:24:25:25
43 pinwheels = 0.139535 24:24:24:25:25:25:25
44 pinwheels = 0.130682 23:24:25:26:26:26:26
45 pinwheels = 0.133333 24:26:26:26:26:26:26
46 pinwheels = 0.141304 26:26:26:26:26:27:27
47 pinwheels = 0.138298 26:27:27:27:27:27:27
48 pinwheels = 0.140625 27:27:27:27:28:28:28
49 pinwheels = 0.127551 25:28:28:28:29:29:29
50 pinwheels = 0.135 27:28:29:29:29:29:29
51 pinwheels = 0.142157 29:29:29:29:29:29:30
52 pinwheels = 0.134615 28:30:30:30:30:30:30
53 pinwheels = 0.141509 30:30:30:30:30:31:31
54 pinwheels = 0.12963 28:28:32:32:32:32:32
55 pinwheels = 0.140909 31:31:31:31:32:32:32
56 pinwheels = 0.142857 32:32:32:32:32:32:32
57 pinwheels = 0.140351 32:32:32:33:33:33:33
58 pinwheels = 0.137931 32:33:33:33:33:34:34
59 pinwheels = 0.139831 33:33:34:34:34:34:34
60 pinwheels = 0.141667 34:34:34:34:34:35:35
61 pinwheels = 0.139344 34:34:34:35:35:36:36
62 pinwheels = 0.141129 35:35:35:35:36:36:36
63 pinwheels = 0.138889 35:36:36:36:36:36:37
64 pinwheels = 0.128906 33:34:37:38:38:38:38
65 pinwheels = 0.138462 36:37:37:37:37:38:38
66 pinwheels = 0.132576 35:36:38:38:39:39:39
67 pinwheels = 0.126866 34:36:39:39:40:40:40
68 pinwheels = 0.139706 38:38:39:39:39:39:40
69 pinwheels = 0.137681 38:39:39:40:40:40:40
70 pinwheels = 0.132143 37:38:41:41:41:41:41
71 pinwheels = 0.137324 39:40:40:41:41:41:42
72 pinwheels = 0.138889 40:40:41:41:42:42:42
73 pinwheels = 0.136986 40:41:42:42:42:42:43
74 pinwheels = 0.138514 41:41:42:43:43:43:43
75 pinwheels = 0.136667 41:41:43:43:44:44:44
76 pinwheels = 0.131579 40:42:44:44:44:45:45
77 pinwheels = 0.13961 43:44:44:44:44:44:45
78 pinwheels = 0.137821 43:44:45:45:45:45:45
79 pinwheels = 0.139241 44:45:45:45:45:46:46
80 pinwheels = 0.140625 45:45:46:46:46:46:46
81 pinwheels = 0.132716 43:45:46:47:47:48:48
82 pinwheels = 0.140244 46:46:47:47:47:47:48
83 pinwheels = 0.138554 46:47:47:48:48:48:48
84 pinwheels = 0.136905 46:46:48:49:49:49:49
85 pinwheels = 0.141176 48:48:48:49:49:49:49
86 pinwheels = 0.130814 45:45:50:51:51:51:51
87 pinwheels = 0.132184 46:48:50:51:51:51:51
88 pinwheels = 0.139205 49:49:50:50:51:51:52
89 pinwheels = 0.13764 49:50:51:51:51:52:52
90 pinwheels = 0.136111 49:50:51:52:52:53:53
91 pinwheels = 0.14011 51:52:52:52:52:52:53
92 pinwheels = 0.138587 51:52:52:53:53:53:54
93 pinwheels = 0.129032 48:49:55:55:55:55:55
94 pinwheels = 0.140957 53:53:53:54:54:54:55
95 pinwheels = 0.139474 53:54:54:54:55:55:55
96 pinwheels = 0.140625 54:54:55:55:55:55:56
97 pinwheels = 0.128866 50:50:56:58:58:58:58
98 pinwheels = 0.137755 54:55:56:56:56:57:58
99 pinwheels = 0.131313 52:53:57:58:58:59:59
100 pinwheels = 0.14 56:56:57:57:57:58:59

[Tom Sirgedas]

maximum density for one color

2  pinwheels = 0.125     (1/8)
3  pinwheels = 0.0833333 (1/12)
4  pinwheels = 0.125     (2/16)
5  pinwheels = 0.15      (3/20)
6  pinwheels = 0.125     (3/24)
7  pinwheels = 0.142857  (4/28)
8  pinwheels = 0.125     (4/32)
9  pinwheels = 0.138889  (5/36)
10 pinwheels = 0.15      (6/40)
11 pinwheels = 0.136364  (6/44)
12 pinwheels = 0.145833  (7/48)
13 pinwheels = 0.134615  (7/52)
14 pinwheels = 0.142857  (8/56)
15 pinwheels = 0.15      (9/60)
16 pinwheels = 0.140625  (9/64)
17 pinwheels = 0.147059  (10/68)
18 pinwheels = 0.138889  (10/72)
19 pinwheels = 0.144737  (11/76)
20 pinwheels = 0.15      (12/80)
21 pinwheels = 0.142857  (12/84)
22 pinwheels = 0.147727  (13/88)
23 pinwheels = 0.141304  (13/92)
24 pinwheels = 0.145833  (14/96)
25 pinwheels = 0.15      (15/100)
26 pinwheels = 0.144231  (15/104)
27 pinwheels = 0.148148  (16/108)
28 pinwheels = 0.142857  (16/112)
29 pinwheels = 0.146552  (17/116)
30 pinwheels = 0.15      (18/120)
31 pinwheels = 0.145161  (18/124)
32 pinwheels = 0.148438  (19/128)
33 pinwheels = 0.143939  (19/132)
34 pinwheels = 0.147059  (20/136)
35 pinwheels = 0.15      (21/140)
36 pinwheels = 0.145833  (21/144)
37 pinwheels = 0.148649  (22/148)
38 pinwheels = 0.144737  (22/152)
39 pinwheels = 0.147436  (23/156)
40 pinwheels = 0.15      (24/160)
41 pinwheels = 0.146341  (24/164)
42 pinwheels = 0.14881   (25/168)
43 pinwheels = 0.145349  (25/172)
44 pinwheels = 0.147727  (26/176)
45 pinwheels = 0.15      (27/180)
46 pinwheels = 0.146739  (27/184)
47 pinwheels = 0.148936  (28/188)
48 pinwheels = 0.145833  (28/192)
49 pinwheels = 0.147959  (29/196)
50 pinwheels = 0.15      (30/200)
51 pinwheels = 0.147059  (30/204)
52 pinwheels = 0.149038  (31/208)
53 pinwheels = 0.146226  (31/212)

[jaan3...@gmail.com]

You get it so fast! One can check also maximum density of non-7-colorable tessellations (which you examined already), if it will take not more time. Aubrey conjectured 1/6 bound for any "necessary" tessellations. So for these tessellations one can check 8-colorings.

[Tom Sirgedas]

"Unit" is the 12 highlighted tiles:

Densest mono-coloring:

Here's the densest mono-coloring I found: 0.13888

It's 6x1 units and each row is shifted over 2 units. This corresponds to 6,0 2,1 in the table below.

1,0 0,1 = 0.0833333  (1/12)

2,0 0,1 = 0.0833333  (2/24)

2,0 1,1 = 0.0833333  (2/24)

3,0 0,1 = 0.111111  (4/36)

3,0 1,1 = 0.111111  (4/36)

3,0 2,1 = 0.111111  (4/36)

4,0 0,1 = 0.104167  (5/48)

4,0 1,1 = 0.104167  (5/48)

4,0 2,1 = 0.104167  (5/48)

4,0 3,1 = 0.104167  (5/48)

5,0 0,1 = 0.1  (6/60)

5,0 1,1 = 0.1  (6/60)

5,0 2,1 = 0.133333  (8/60)

5,0 3,1 = 0.133333  (8/60)

5,0 4,1 = 0.133333  (8/60)

6,0 0,1 = 0.111111  (8/72)

6,0 1,1 = 0.111111  (8/72)

6,0 2,1 = 0.138889  (10/72)

6,0 3,1 = 0.111111  (8/72)

6,0 4,1 = 0.111111  (8/72)

6,0 5,1 = 0.138889  (10/72)

7,0 0,1 = 0.107143  (9/84)

7,0 1,1 = 0.107143  (9/84)

7,0 2,1 = 0.130952  (11/84)

7,0 3,1 = 0.130952  (11/84)

7,0 4,1 = 0.130952  (11/84)

7,0 5,1 = 0.130952  (11/84)

7,0 6,1 = 0.130952  (11/84)
...

20,0 17,1 = 0.133333  (32/240)

20,0 18,1 = 0.133333  (32/240)

20,0 19,1 = 0.133333  (32/240)

[Tom Sirgedas]

The 10-tile unit can be mono-colored with a density of .14444, despite not having a 7-coloring.

This is with 18 units with a shift of 6 units. I only checked up to 19 units, I'll search a little deeper for an even denser coloring.

[jaan3...@gmail.com]

Thank you, excellent!