= 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 \b[ready-to-use randomized <row-column design>] is:
|| Volunteer
|| Wednesday 1
|| Wednesday 2
|| Wednesday 3
|| Wednesday 4
|| Wednesday 5
|| Wednesday 6
| 1
| A
| B
| C
| A
| B
| C
| 2
| C
| A
| B
| C
| A
| B
| 3
| B
| C
| A
| B
| C
| A
| 4
| C
| B
| A
| C
| B
| A
| 5
| A
| C
| B
| A
| C
| B
| 6
| B
| A
| C
| B
| A
| C
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.
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