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
none 		125
?
6
none 		16
none 		26
none 		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
none 		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
none 		138
none 		238
none 		1238
none
48
none 		148
none 		248
none 		1248
none
348
none 		1348
none 		2348
none 		12348
none
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.