# 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:

 4none 5none 15? 125? 6none 16none 26? 126? 136? 1236? (Johannes Waldmann) 56none 156? (Johannes Waldmann) (Johannes Waldmann) (George Sicherman) (George Sicherman) (Maurizio Morandi) 7none 17none 27none 127none 37? 137? 237? 1237? 147? (Maurizio Morandi) 1247? (Johannes Waldmann) 57none 157? (Johannes Waldmann) 1257? (Maurizio Morandi) (Maurizio Morandi) (Maurizio Morandi) (Maurizio Morandi) 67none 167none 267? 1267? (Maurizio Morandi) 1367? (Johannes Waldmann) 12367? (Johannes Waldmann) (Johannes Waldmann) (Maurizio Morandi) 567none 1567? (George Sicherman) (Johannes Waldmann) (George Sicherman) (Johannes Waldmann) (Maurizio Morandi) (George Sicherman) 8none 18none 28none 128none 38? 138? 238? 1238? 48? 148? 248? 1248? 348? 1348? 2348? 12348? 58none 158? 258? 1258? 358? 1358? 2358? 12358? (Johannes Waldmann) (Johannes Waldmann) (Johannes Waldmann) (Maurizio Morandi) (Maurizio Morandi) (Johannes Waldmann) (Johannes Waldmann) (Johannes Waldmann) 68none 168none 268? 1268? (Johannes Waldmann) 1368? (Maurizio Morandi) 12368? 468? (Johannes Waldmann) (Maurizio Morandi) (Johannes Waldmann) (Maurizio Morandi) (Johannes Waldmann) (Maurizio Morandi) (Maurizio Morandi) 568none 1568? 12568? (Maurizio Morandi) (Maurizio Morandi) (Maurizio Morandi) (Maurizio Morandi) 78none 178none 278none 1278none 378? 1378? 2378? 12378? 478? 1478? (Johannes Waldmann) 12478? (Maurizio Morandi) (Johannes Waldmann) (Johannes Waldmann) (Johannes Waldmann) 578none 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) 678none 1678none 2678? 12678? (Johannes Waldmann) 13678? (Maurizio Morandi) 123678? (Johannes Waldmann) (Johannes Waldmann) (Johannes Waldmann) (Maurizio Morandi) 5678none (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:

 3none 4none 14none 34none

George Sicherman also sent these polyhex solutions:

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

George Sicherman also sent these polyiamond solutions:

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

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

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

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

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