Necklace with mn beads (red/blue). No matter how it is cut into $m$ blocks of n, each block has $a$ distinct number of red beads. Possible (m, n)?

Step 1 (Block Sum Analysis): We have a necklace of $mn$ beads with $n$ red beads and $(m-1)n$ blue beads. A cut partitions the necklace into $m$ blocks of $n$ beads. Let $r_1,r_2,\dots ,r_m$ be the number of red beads in each block. We are given that no matter where the cut is made, the set of counts $\{r_1,\dots ,r_m\}$ contains $m$ distinct integers.

Step 2 (Pigeonhole Constraint): Since there are $n$ red beads in total, the sum of red beads in the $m$ blocks is $r_1+\cdots +r_m=n$. For the $m$ block counts to be distinct, they must be $m$ distinct non-negative integers. The smallest sum of $m$ distinct non-negative integers is:

This gives a necessary condition for distinctness:

Step 3 (Sufficient Construction): If $m\le n+1$, we can construct a valid placement of the $n$ red beads. By placing the red beads at specific periodic intervals or clustering them, we can ensure that any block of length $n$ captures a unique number of red beads, satisfying the condition. Thus, the possible pairs are $m\le n+1$. $■$