Is There a Strongest Die in a Set of Dice with the Same Mean Pips?

Jan-ken, a.k.a. rock-paper-scissors, is a cerebrated example of non-transitive game with three (pure) strategies, rock, paper and scissors. Interestingly, any Jan-ken generalized to four strategies contains at least one useless strategy unless it allows tie between distinct pure strategies. Non-transitive dice could be stochastic analogue Jan-ken: the transitivity does not hold on some sets dice, e.g., Efron's dice. Including this interested in which do contain die. In particular, we are concerned existence strongest (or weakest, symmetrically) die set under two conditions that (1) number appears most side, i.e., no break (2) mean pips same. We firstly prove never exist if n m-sided given as partition numbers {1,…,mn}. Next, show sufficient exists numbers. also give algorithms find includes