What is the minimum number of dice painted with one letter on all six sides such that all permutations...

combinatorics/alphabet.blocks

Problem

What is the minimum number of dice painted with one letter on all six sides
such that all permutations without repetitions of n letters can be formed
by placing n dice together in a line?

Show solution

n=   2          3          4          5          6
   (8,4)      (9,7)      (9,3)      (10,7)     (11,7)

   aijklm     abcde?     acdefg     abcde?     abkmuz
   bijklm     fghij?     bhijkl     fghij?     bcpwy?
   cnopqr     klmno?     cmnopq     klmno?     cdlnvz
   dnopqr     pqrst?     dhnrvy     pqrst?     deqxu?
   estuvw     uvwxy?     eiosvw     uvwxy?     efmowz
   fstuvw     afkpu?     fjptwx     afkpu?     fgryv?
   gxyz??     bglqv?     gkquxy     bglqv?     ghnkx?
   hxyz??     chmrwz     lmrstu     chmrwz     hisuw?
              dinsxz     zab???     dinsxz     ijoly?
                                    ejotyz     jatvx?
                                               pqrstz

I think I can prove that there is no solution with 11 dice with 9
don't cares or with 10 dice, but I haven't checked all the details, so
I might have made a mistake.  In any case, that leaves open the case
of 11 dice with 8 don't cares; my guess is that it is not possible.

-- John Rickard (jrickard@eoe.co.uk)