Problem 6

Draw the broken path through consecutive cells of $C$, joining its first cell to the southwest corner and its last cell to the northeast corner. Draw the analogous path through $A$, joining its first and last cells to the northwest and southeast corners. Because $C$ is a chain and $A$ an antichain, the two paths do not cross except at their common cell $c$; they divide the board into north, east, south, and west regions. For every black cell in the north region, write $N$ in the cell immediately above it; similarly write $S$ below cells in the south region, $W$ to the left in the west region, and $E$ to the right in the east region. A cell on a path receives every direction belonging to the adjacent regions. Since $c$ belongs to both paths, it receives all four letters; every other cell of $A\cup C$ receives two letters, and every cell outside $A\cup C$ receives one. Hence the total number of letters is $C=n+a+b+1$ (the notation $C$ here denotes the letter count, not the chain).