perpetual calendar on five dice

Question

Does anyone know how to create a perpetual calendar on the sides of five dice ? The calendar is required to show just two digit day, and three letter month name. Apparently this can be done using French month names - i.e. (JAN FEV MAR AVR MAI JUN JUI AOU SEP OCT NOV DEC) but not possible using English three letter month names. I have tried this, and it just seems a misery of endless tinkering. Is there a method in it somewhere ?

Answer 1

For the English three letter abbreviations of the months we need 19 different letters (only H, I, K, Q, W, X, Z are not needed), one more than the $18$ available faces on three dice. Maybe we could cheat a little, as with $6$ and $9$ on the digits dice. But there is another point: The graph $\Gamma$ containing the $19$ occurring letters as vertices, and an edge for any two letters that should not appear on the same die, contains a $K_4$ consisting of A, J, N, and U; see the figure below. Therefore we need at least four dice to deal with the letters of such a calendar. A way out could be the following: Introduce a sixth die allowing for a slash: JAN/15. Now the problem becomes again interesting, and it is not too difficult to devise a solution. Of course we have to cheat a little and use the $6$ for a $9$ as well. The six dice could show the following entries:

(0,1,2,3,4,5), (0,1,2,6,7,8),

(A,E,T,V,/,*), (B,M,P,O,U,/), (C,L,N,R,S,/), (D,F,G,J,Y,/).

The * denotes a free space for the logo of the manufacturer. The four-coloring of $\Gamma$ shown in the following figure was set up in such a way that no color occurs more than five times, allowing for a slash on each letter-die.