 Solution

Start with six program sequences: $ABCABC$, $ACBACB$, $BACBAC$, $BCABCA$, $CABCAB$ and $CBACBA$. This gives two appearances of each program per row and per column. It also balances transitions: each ordered pair of distinct programs occurs five times across all adjacent sessions.

Apply restricted randomization by randomly assigning the six sequences to the six volunteers and independently permuting the three program labels. Using the supplied numbers, rank the first six from smallest to largest: the sequence indices are $(4,1,5,2,3,6)$. Assign these, in order, to volunteers 1 through 6. Rank the next three numbers attached to symbols $(A,B,C)$: their order is $(B,C,A)$. Assign actual program labels $(A,B,C)$ to these ordered symbols, so base $A\mapsto C$, base $B\mapsto A$, base $C\mapsto B$. The final ready-to-use randomized row-column design is:

Programs retain their real identities after this label permutation. The researcher should follow the table across chronological afternoons; arbitrary column permutations are deliberately excluded so that transition balance survives. Each program receives twelve sessions and occurs twice in every block. Under the additive block model, the analysis of variance has row, column, program and residual statistical degrees of freedom $5,5,2,23$, respectively, besides the grand mean. Transition balance is useful against differential carryover effects, but is not a proof that they are absent.