Talk:Henry Ernest Dudeney/Puzzles and Curious Problems/360 - Fort Garrisons/Solution

There are 120 solutions.

1	[16] [18, 24, 36, 22] [28, 20] [26] [32, 28]
2	[16] [18, 28, 28, 26] [24, 20] [22] [32, 36]
3	[16] [18, 28, 28, 26] [36, 32] [22] [20, 24]
4	[16] [18, 36, 24, 22] [28, 32] [26] [20, 28]
5	[16] [22, 20, 32, 26] [36, 28] [18] [28, 24]
6	[16] [22, 24, 36, 18] [32, 28] [26] [28, 20]
7	[16] [22, 32, 20, 26] [24, 28] [18] [28, 36]
8	[16] [22, 36, 24, 18] [20, 28] [26] [28, 32]
9	[16] [26, 20, 32, 22] [28, 24] [18] [36, 28]
10	[16] [26, 28, 28, 18] [20, 24] [22] [36, 32]
11	[16] [26, 28, 28, 18] [32, 36] [22] [24, 20]
12	[16] [26, 32, 20, 22] [28, 36] [18] [24, 28]
13	[18] [16, 24, 28, 32] [36, 26] [20] [22, 28]
14	[18] [16, 28, 24, 32] [28, 22] [20] [26, 36]
15	[18] [16, 28, 36, 20] [28, 22] [32] [26, 24]
16	[18] [16, 36, 28, 20] [24, 26] [32] [22, 28]
17	[18] [20, 22, 26, 32] [36, 28] [16] [24, 28]
18	[18] [20, 26, 22, 32] [28, 24] [16] [28, 36]
19	[18] [20, 28, 36, 16] [26, 24] [32] [28, 22]
20	[18] [20, 36, 28, 16] [22, 28] [32] [24, 26]
21	[18] [32, 22, 26, 20] [24, 28] [16] [36, 28]
22	[18] [32, 24, 28, 16] [22, 28] [20] [36, 26]
23	[18] [32, 26, 22, 20] [28, 36] [16] [28, 24]
24	[18] [32, 28, 24, 16] [26, 36] [20] [28, 22]
25	[20] [18, 22, 36, 24] [26, 16] [28] [32, 28]
26	[20] [18, 26, 28, 28] [22, 16] [24] [32, 36]
27	[20] [18, 28, 26, 28] [36, 32] [24] [16, 22]
28	[20] [18, 36, 22, 24] [28, 32] [28] [16, 26]
29	[20] [24, 16, 32, 28] [36, 26] [18] [28, 22]
30	[20] [24, 22, 36, 18] [32, 28] [28] [26, 16]
31	[20] [24, 32, 16, 28] [22, 28] [18] [26, 36]
32	[20] [24, 36, 22, 18] [16, 26] [28] [28, 32]
33	[20] [28, 16, 32, 24] [28, 22] [18] [36, 26]
34	[20] [28, 26, 28, 18] [32, 36] [24] [22, 16]
35	[20] [28, 28, 26, 18] [16, 22] [24] [36, 32]
36	[20] [28, 32, 16, 24] [26, 36] [18] [22, 28]
37	[22] [16, 20, 36, 28] [32, 18] [28] [26, 24]
38	[22] [16, 24, 32, 28] [36, 26] [28] [18, 20]
39	[22] [16, 32, 24, 28] [20, 18] [28] [26, 36]
40	[22] [16, 36, 20, 28] [24, 26] [28] [18, 32]
41	[22] [28, 18, 26, 28] [24, 20] [16] [36, 32]
42	[22] [28, 18, 26, 28] [36, 32] [16] [24, 20]
43	[22] [28, 20, 36, 16] [26, 24] [28] [32, 18]
44	[22] [28, 24, 32, 16] [18, 20] [28] [36, 26]
45	[22] [28, 26, 18, 28] [20, 24] [16] [32, 36]
46	[22] [28, 26, 18, 28] [32, 36] [16] [20, 24]
47	[22] [28, 32, 24, 16] [26, 36] [28] [20, 18]
48	[22] [28, 36, 20, 16] [18, 32] [28] [24, 26]
49	[24] [20, 16, 36, 28] [32, 18] [26] [28, 22]
50	[24] [20, 22, 32, 26] [36, 28] [28] [18, 16]
51	[24] [20, 32, 22, 26] [16, 18] [28] [28, 36]
52	[24] [20, 36, 16, 28] [22, 28] [26] [18, 32]
53	[24] [26, 18, 28, 28] [22, 16] [20] [36, 32]
54	[24] [26, 22, 32, 20] [18, 16] [28] [36, 28]
55	[24] [26, 28, 18, 28] [32, 36] [20] [16, 22]
56	[24] [26, 32, 22, 20] [28, 36] [28] [16, 18]
57	[24] [28, 16, 36, 20] [28, 22] [26] [32, 18]
58	[24] [28, 18, 28, 26] [36, 32] [20] [22, 16]
59	[24] [28, 28, 18, 26] [16, 22] [20] [32, 36]
60	[24] [28, 36, 16, 20] [18, 32] [26] [22, 28]
61	[26] [16, 20, 28, 36] [32, 18] [24] [22, 28]
62	[26] [16, 28, 20, 36] [28, 22] [24] [18, 32]
63	[26] [16, 28, 32, 24] [28, 22] [36] [18, 20]
64	[26] [16, 32, 28, 24] [20, 18] [36] [22, 28]
65	[26] [24, 18, 22, 36] [28, 20] [16] [28, 32]
66	[26] [24, 22, 18, 36] [32, 28] [16] [20, 28]
67	[26] [24, 28, 32, 16] [18, 20] [36] [28, 22]
68	[26] [24, 32, 28, 16] [22, 28] [36] [20, 18]
69	[26] [36, 18, 22, 24] [28, 32] [16] [28, 20]
70	[26] [36, 20, 28, 16] [22, 28] [24] [32, 18]
71	[26] [36, 22, 18, 24] [20, 28] [16] [32, 28]
72	[26] [36, 28, 20, 16] [18, 32] [24] [28, 22]
73	[28] [20, 16, 28, 36] [32, 18] [22] [24, 26]
74	[28] [20, 26, 32, 22] [28, 24] [36] [18, 16]
75	[28] [20, 28, 16, 36] [26, 24] [22] [18, 32]
76	[28] [20, 32, 26, 22] [16, 18] [36] [24, 28]
77	[28] [22, 18, 24, 36] [26, 16] [20] [28, 32]
78	[28] [22, 18, 36, 24] [26, 16] [32] [28, 20]
79	[28] [22, 20, 26, 32] [36, 28] [24] [16, 18]
80	[28] [22, 24, 18, 36] [32, 28] [20] [16, 26]
81	[28] [22, 26, 20, 32] [18, 16] [24] [28, 36]
82	[28] [22, 26, 32, 20] [18, 16] [36] [28, 24]
83	[28] [22, 32, 26, 20] [24, 28] [36] [16, 18]
84	[28] [22, 36, 18, 24] [20, 28] [32] [16, 26]
85	[28] [24, 16, 28, 32] [36, 26] [22] [20, 18]
86	[28] [24, 18, 36, 22] [28, 20] [32] [26, 16]
87	[28] [24, 28, 16, 32] [18, 20] [22] [26, 36]
88	[28] [24, 36, 18, 22] [16, 26] [32] [20, 28]
89	[28] [32, 16, 28, 24] [20, 18] [22] [36, 26]
90	[28] [32, 20, 26, 22] [16, 18] [24] [36, 28]
91	[28] [32, 26, 20, 22] [28, 36] [24] [18, 16]
92	[28] [32, 28, 16, 24] [26, 36] [22] [18, 20]
93	[28] [36, 16, 28, 20] [24, 26] [22] [32, 18]
94	[28] [36, 18, 24, 22] [28, 32] [20] [26, 16]
95	[28] [36, 24, 18, 22] [16, 26] [20] [32, 28]
96	[28] [36, 28, 16, 20] [18, 32] [22] [26, 24]
97	[32] [18, 22, 24, 36] [26, 16] [28] [20, 28]
98	[32] [18, 24, 22, 36] [28, 20] [28] [16, 26]
99	[32] [18, 26, 28, 28] [22, 16] [36] [20, 24]
100	[32] [18, 28, 26, 28] [24, 20] [36] [16, 22]
101	[32] [28, 16, 20, 36] [28, 22] [18] [24, 26]
102	[32] [28, 20, 16, 36] [26, 24] [18] [22, 28]
103	[32] [28, 26, 28, 18] [20, 24] [36] [22, 16]
104	[32] [28, 28, 26, 18] [16, 22] [36] [24, 20]
105	[32] [36, 16, 20, 28] [24, 26] [18] [28, 22]
106	[32] [36, 20, 16, 28] [22, 28] [18] [26, 24]
107	[32] [36, 22, 24, 18] [20, 28] [28] [26, 16]
108	[32] [36, 24, 22, 18] [16, 26] [28] [28, 20]
109	[36] [26, 18, 28, 28] [22, 16] [32] [24, 20]
110	[36] [26, 20, 22, 32] [28, 24] [28] [16, 18]
111	[36] [26, 22, 20, 32] [18, 16] [28] [24, 28]
112	[36] [26, 28, 18, 28] [20, 24] [32] [16, 22]
113	[36] [28, 16, 24, 32] [28, 22] [26] [20, 18]
114	[36] [28, 18, 28, 26] [24, 20] [32] [22, 16]
115	[36] [28, 24, 16, 32] [18, 20] [26] [22, 28]
116	[36] [28, 28, 18, 26] [16, 22] [32] [20, 24]
117	[36] [32, 16, 24, 28] [20, 18] [26] [28, 22]
118	[36] [32, 20, 22, 26] [16, 18] [28] [28, 24]
119	[36] [32, 22, 20, 26] [24, 28] [28] [18, 16]
120	[36] [32, 24, 16, 28] [22, 28] [26] [18, 20]

--Julia Hartman (talk) 17:44, 4 June 2023 (UTC)

Superb, feel free to post this up as a page offering this final solution, complete with the algorithm of the program that got there (if you get so moved).

I believe this should admit a considerable simplification. This is a regular star, so what we can do already is reduce the number of solutions by a factor of 5 for each tip of the star and then by a factor of 2 due to left-right symmetry. So really we need only 12 unique solutions, and the rest can be generated thence. But of course, formally these all are solutions.--Julius (talk) 16:57, 5 June 2023 (UTC)

You are right: 12 unique solutions. Works for me.

1	[16] [18, 24, 36, 22] [28, 20] [26] [32, 28]
2	[16] [18, 28, 28, 26] [24, 20] [22] [32, 36]
3	[16] [18, 28, 28, 26] [36, 32] [22] [20, 24]
4	[16] [18, 36, 24, 22] [28, 32] [26] [20, 28]
5	[16] [22, 20, 32, 26] [36, 28] [18] [28, 24]
7	[16] [22, 32, 20, 26] [24, 28] [18] [28, 36]
17	[18] [20, 22, 26, 32] [36, 28] [16] [24, 28]
18	[18] [20, 26, 22, 32] [28, 24] [16] [28, 36]
29	[20] [24, 16, 32, 28] [36, 26] [18] [28, 22]
31	[20] [24, 32, 16, 28] [22, 28] [18] [26, 36]
41	[22] [28, 18, 26, 28] [24, 20] [16] [36, 32]
53	[24] [26, 18, 28, 28] [22, 16] [20] [36, 32]

--Julia Hartman (talk) 12:32, 8 June 2023 (UTC)

Solutions and classes: For example, solution #24 is a rotated version of solution #1; solution #24 is also a symmetric version of solution #13.

[1, 24, 40, 79, 103]	 -- [8, 13, 43, 91, 100]
[2, 22, 62, 107, 110]	 -- [10, 14, 70, 98, 119]
[3, 20, 30, 56, 63]	 -- [11, 15, 28, 50, 68]
[4, 19, 34, 38, 83]	 -- [6, 16, 27, 47, 74]
[5, 48, 55, 64, 86]	 -- [12, 37, 58, 67, 84]
[7, 44, 61, 94, 112]	 -- [9, 39, 72, 80, 114]
[17, 32, 57, 92, 99]	 -- [23, 25, 52, 85, 104]
[18, 35, 93, 97, 120]	 -- [21, 26, 75, 108, 113]
[29, 46, 60, 76, 78]	 -- [36, 42, 49, 82, 88]
[31, 54, 69, 73, 116]	 -- [33, 51, 66, 96, 109]
[41, 81, 95, 102, 117]	 -- [45, 90, 77, 105, 115]
[53, 71, 87, 101, 118]	 -- [59, 65, 89, 106, 111]

--Julia Hartman (talk) 11:59, 14 June 2023 (UTC)