Problem of the Month (December 2016)

For a configuration of unit squares on a grid, let the adjacency number of a square be the number of (horizontally, vertically, or diagonally) adjacent squares. We call a connected configuration of squares equally adjacent if every number that is an adjacency number for the configuration has an equal number of squares with that adjacency number. For example, here are the smallest known equally adjacent configurations for the non-empty subsets of {1,2,3}:

What subsets of {1,2,3,4,5,6,7,8} have equally adjacent configurations? What is the smallest configuration representing each one? What are the solutions for the hexagonal grid? There are too many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} to fully consider solutions on the triangular grid, but what about the one-element and two-element subsets?


ANSWERS

Johannes Waldmann, George Sicherman, and Maurizio Morandi sent improvements.

The smallest known configurations are shown below:

4
none
5
none
15
?
125
?
6
none
16
none
26
?
126
?
136
?
1236
?

(Johannes Waldmann)
56
none
156
?

(Johannes Waldmann)

(Johannes Waldmann)

(George Sicherman)

(George Sicherman)

(Maurizio Morandi)
7
none
17
none
27
none
127
none
37
?
137
?
237
?
1237
?
147
?

(Maurizio Morandi)
1247
?

(Johannes Waldmann)
57
none
157
?

(Johannes Waldmann)
1257
?

(Maurizio Morandi)

(Maurizio Morandi)

(Maurizio Morandi)

(Maurizio Morandi)
67
none
167
none
267
?
1267
?

(Maurizio Morandi)
1367
?

(Johannes Waldmann)
12367
?

(Johannes Waldmann)

(Johannes Waldmann)

(Maurizio Morandi)
567
none
1567
?

(George Sicherman)

(Johannes Waldmann)

(George Sicherman)

(Johannes Waldmann)

(Maurizio Morandi)

(George Sicherman)
8
none
18
none
28
none
128
none
38
?
138
?
238
?
1238
?
48
?
148
?
248
?
1248
?
348
?
1348
?
2348
?
12348
?
58
none
158
?
258
?
1258
?
358
?
1358
?
2358
?
12358
?

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(Maurizio Morandi)

(Maurizio Morandi)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)
68
none
168
none
268
?
1268
?

(Johannes Waldmann)
1368
?

(Maurizio Morandi)
12368
?
468
?

(Johannes Waldmann)

(Maurizio Morandi)

(Johannes Waldmann)

(Maurizio Morandi)

(Johannes Waldmann)

(Maurizio Morandi)

(Maurizio Morandi)
568
none
1568
?
12568
?

(Maurizio Morandi)

(Maurizio Morandi)

(Maurizio Morandi)

(Maurizio Morandi)
78
none
178
none
278
none
1278
none
378
?
1378
?
2378
?
12378
?
478
?
1478
?

(Johannes Waldmann)
12478
?

(Maurizio Morandi)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)
578
none
1578
?

(Maurizio Morandi)
12578
?

(Johannes Waldmann)

(Maurizio Morandi)

(Johannes Waldmann)

(Maurizio Morandi)

(Maurizio Morandi)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(Maurizio Morandi)

(Maurizio Morandi)

(Maurizio Morandi)
678
none
1678
none
2678
?
12678
?

(Johannes Waldmann)
13678
?

(Maurizio Morandi)
123678
?

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(Maurizio Morandi)
5678
none

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

(George Sicherman)

(Johannes Waldmann)

(Johannes Waldmann)

(Johannes Waldmann)

George Sicherman sent these solutions where squares have only 4 neighbors:

3
none
4
none
14
none
34
none

George Sicherman also sent these polyhex solutions:

4
none
14
?
5
none
15
none
25
?
125
?
135
?
1235
?
45
none
145
?
6
none
16
none
26
none
126
none
36
?
136
?
236
?
1236
?
46
none
146
?
246
?
1246
?
1346
?
2346
?
12346
?
56
none
156
none
256
?
1256
?
1356
?
2356
?
12356
?
456
none
1456
?
12456
?
13456
?

George Sicherman also sent these polyiamond solutions:

15
none
16
none
56
?
6
?
17
none
47
?
67
?
18
none
28
?
38
?
48
?
68
?
19
none
29
?
39
?
49
?
59
?
69
?
1 10
none
2 10
none
3 10
?
4 10
?
5 10
?
6 10
?
1 11
none
2 11
none
3 11
none
4 11
?
5 11
?
6 11
?
1 12
none
2 12
none
3 12
none
4 12
none
5 12
?
6 12
?

George Sicherman also sent these polycairo solutions, with edge connections:

3
?
4
none
14
?
34
?
5
none
15
none
25
?
125
?
35
?
135
?
1235
?
45
none
145
?
245
?
345
?
1345
?
12345
?

George Sicherman also sent these polycairo solutions, with vertex connections:

4
?
5
none
15
?
125
?
45
none
6
none
16
none
26
?
126
?
36
?
136
?
1236
?
46
none
146
?
56
none
156
?
256
?
1256
?
1356
?
456
none
1456
?

       

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/19/16.