Combinatorics · Symmetry and distinguishability · Cyclic orders · Configurations of rings and keys · Extremal construction
A bank guard looks after \(n\) safes. Every safe has its own key, no key fits two safes, and all the keys look alike. He is given as many identical circular metal rings as he wants. On any ring he may hook any number of keys and any number of other rings, with one restriction: two rings may never hook on one and the same key. The rings are such that, once he has hooked everything he wants to hook, the cyclic order of the objects on each ring can no longer be disturbed.
He wants to join all the keys into one single object in such a way that from then on he can always tell, from the arrangement of keys and rings alone and without trying any key in a lock, which key opens a given safe. For a given \(n>1\), find the least number of rings he needs if
(a) every key has exactly one axis of symmetry in the plane of the key;
(b) the keys have no axis of symmetry.
Sign in to check answers, open hints, read the full solution, and track your progress. Statements are always free.
Serbian National Competition (Drzavno takmicenje) 2015, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source