IMO 2017 Shortlist C6
Let n ą 1 be an integer. An n ˆ n ˆ n ube is omposed of n3 unit ubes. Ea h unit ube is painted with one olor. For ea h n...
Category: Combinatorics
Problem
Let n ą 1 be an integer. An n ˆ n ˆ n ube is omposed of n3 unit ubes. Ea h unit ube is painted with one olor. For ea h nˆnˆ1 box onsisting of n2 unit ubes (of any of the three possible orientations), we onsider the set of the olors present in that box (ea h olor is listed only on e). This way, we get 3n sets of olors, split into three groups a ording to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of n, the maximal possible number of olors that are present. (Russia)