# Problem of the Month (August 2017)

For a set S of U.S. states, we say its adjacency set A(S) is the set of numbers that count how many states in S are adjacent to each state in S. (You can find the neighboring states here except we do not consider the point contacts of Colorado-Arizona, New Mexico-Utah, or the water connections of Michigan-Minnesota, Michigan-Illinois, or New York-Rhode Island neighbors.) If T is a non-empty subset of {1,2,3,4,5,6,7,8}, does there exist a collection of states S with A(S)=T? If so, what is the largest such set? Which largest sets are unique?

Here are the best known solutions.

One Digit
 {1}: 24 states {2}: 27 states (George Sicherman) {3}: none {4}: none {5}: none {6}: none {7}: none {8}: none

Two Digits
 {1,2}: 31 states (George Sicherman) {1,3}: 30 states (George Sicherman) {1,4}: 23 states {1,5}: none (Bryce Herdt) {1,6}: none {1,7}: none {1,8}: none {2,3}: 34 states (Bryce Herdt) {2,4}: 32 states (Rob Pratt) {2,5}: 23 states {2,6}: none {2,7}: none {2,8}: none {3,4}: 25 states (George Sicherman) {3,5}: 20 states (George Sicherman) {3,6}: 13 states (George Sicherman) {3,7}: 8 states {3,8}: 9 states {4,5}: none (Bryce Herdt) {4,6}: none (Bryce Herdt) {4,7}: none (Bryce Herdt) {4,8}: none (Bryce Herdt) {5,6}: none (Bryce Herdt) {5,7}: none (Bryce Herdt) {5,8}: none {6,7}: none {6,8}: none {7,8}: none

Three Digits
 {1,2,3}: 36 states {1,2,4}: 35 states (Rob Pratt) {1,2,5}: 29 states (Bryce Herdt) {1,2,6}: none {1,2,7}: none {1,2,8}: none {1,3,4}: 36 states (Bryce Herdt) {1,3,5}: 34 states (Rob Pratt) {1,3,6}: 31 states (Bryce Herdt) {1,3,7}: 28 states {1,3,8}: 25 states {1,4,5}: 23 states (Rob Pratt) {1,4,6}: 26 states (George Sicherman) {1,4,7}: none (Rob Pratt) {1,4,8}: none {1,5,6}: none (Rob Pratt) {1,5,7}: none (Rob Pratt) {1,5,8}: none {1,6,7}: none (Rob Pratt) {1,6,8}: none {1,7,8}: none (George Sicherman) {2,3,4}: 39 states (Bryce Herdt) {2,3,5}: 37 states (Rob Pratt) {2,3,6}: 33 states (Rob Pratt) {2,3,7}: 31 states (Bryce Herdt) {2,3,8}: 26 states (Bryce Herdt) {2,4,5}: 36 states (Bryce Herdt) {2,4,6}: 35 states (George Sicherman) {2,4,7}: 31 states (Rob Pratt) {2,4,8}: 28 states (George Sicherman) {2,5,6}: 23 states (Bryce Herdt) {2,5,7}: none (Rob Pratt) {2,5,8}: none {2,6,7}: none (Rob Pratt) {2,6,8}: none {2,7,8}: none (George Sicherman) {3,4,5}: 30 states (Bryce Herdt) {3,4,6}: 29 states (George Sicherman) {3,4,7}: 29 states (Rob Pratt) {3,4,8}: 15 states (George Sicherman) {3,5,6}: 23 states (Rob Pratt) {3,5,7}: 14 states (George Sicherman) {3,5,8}: 23 states (George Sicherman) {3,6,7}: none (Rob Pratt) {3,6,8}: 15 states (George Sicherman) {3,7,8}: none (Bryce Herdt) {4,5,6}: none (Bryce Herdt) {4,5,7}: none (Bryce Herdt) {4,5,8}: none (Bryce Herdt) {4,6,7}: none (Bryce Herdt) {4,6,8}: none (Bryce Herdt) {4,7,8}: none (Bryce Herdt) {5,6,7}: none (Bryce Herdt) {5,6,8}: none (Bryce Herdt) {5,7,8}: none (Bryce Herdt) {6,7,8}: none (Bryce Herdt)

The adjacency sets of the 13 countries of South America, the 16 states of Germany, the the 21 counties of New Jersey, and the the 23 counties of Wyoming can be analyzed completely due to the relatively small number of regions.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/27/17.