Euler's
36 officers problem has been in the news recently, following the passing of the last of 'Euler's Spoilers'.
It notoriously has no solution for the 6x6 square with 6 officers (colonel, lieutenant-colonel, major, captain, lieutenant, and sub-lieutenant) and 6 regiments.
One possible modification, while still keeping 6 regiments, would be to allow substitution of a junior officer (or officers) in one or more of the regiments by a new rank - say, Sensitivity Counsellor, or SC - more reflective of the needs and aspirations of a modern-day military.
What is the minimum number of SC's needed to make the problem solvable?
(In reply to
thought problem about this problem.... by Steven Lord)
Nice work so far Steve! My thoughts on your question:
Lets assume there is a solution with one SC. Any possible solution is part of a family formed by row and column permutations. So we can permute rows 2-6 to guarantee that SC is in row 2 without any issues.
Then permute columns to bring SC into column 1. We may need to relabel AA-FF to keep the strict 1_AA 2_BB 3_CC 4_DD 5_EE 6_FF in toe top row but this should be entirely cosmetic. Renumbering regiment 1-6 is mostly cosmetic - except for the fact that the SC will have a specific number which will be changed if was not already in column 1.
From this I conclude you can place the first SC immediately in row 2 column 1, iterating through regiments 2-6 as possible regiment assignments.
re: thought problem about this problem....
