| [ilmath]
\begin{xy}
\xymatrix{
\ & 0 & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{a} & \mathrm{b} & \mathrm{b} & \mathrm{a} & \mathrm{c} \\
0 & \text{S} \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[d] \ar[r] & \cdot \ar[d] \\
\mathrm{a} & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[d] \ar[r] & \cdot \ar[d]\\
\mathrm{b} & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[d] \ar[r] & \cdot \ar[d]\\
\mathrm{b} & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[d] \ar[r] & \cdot \ar[d]\\
\mathrm{a} & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[r] \ar[d] & \cdot \ar[d] \ar[r] & \cdot \ar[d]\\
\mathrm{b} & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \cdot \ar[r] & \times &
}
% the chains
\ar@{->} "2,2";"3,3"
\ar@{->} "3,3";"4,4"
\ar@{->} "4,3";"5,4"
\ar@{->} "5,2";"6,3"
\ar@{->} "6,3";"7,4"
% first 2 cols done
\ar@{->} "5,5";"6,6"
\ar@{->} "6,6";"7,7"
\ar@{->} "6,7";"7,8"
% bottom 2-chain and 1 chain done
\ar@{->} "2,5";"3,6"
\ar@{->} "3,6";"4,7"
\ar@{->} "4,7";"5,8"
\ar@{->} "5,8";"6,9"
% 4 chain done!
\ar@{->} "4,6";"5,7"
\ar@{->} "2,8";"3,9"
\ar@{->} "3,7";"4,8"
% and the 3 that go / through the 4 chain (the bottom right already being a part of a 2 chain
%START OF RED PATH
\ar@[red]@/^/@{->} "2,2";"2,3"
\ar@[red]@/^/@{->} "2,3";"2,4"
\ar@[red]@/^/@{->} "2,4";"2,5"
\ar@[red]@/^/@{->} "2,5";"2,6"
\ar@[red]@/^/@{->} "2,6";"2,7"
\ar@[red]@/^/@{->} "2,7";"2,8"
\ar@[red]@/^/@{->} "2,8";"2,9"
\ar@[red]@/^/@{->} "2,9";"2,10"
%and down
\ar@[red]@/^/@{->} "2,10";"3,10"
\ar@[red]@/^/@{->} "3,10";"4,10"
\ar@[red]@/^/@{->} "4,10";"5,10"
\ar@[red]@/^/@{->} "5,10";"6,10"
\ar@[red]@/^/@{->} "6,10";"7,10"
%start of green path
\ar@[green]@/_/@{->} "2,2";"2,3"
\ar@[green]@/_/@{->} "2,3";"2,4"
\ar@[green]@/_/@{->} "2,4";"2,5"
%now go down the 4-chain
\ar@[green]@/_/@{->} "2,5";"3,6"
\ar@[green]@/_/@{->} "3,6";"4,7"
\ar@[green]@/_/@{->} "4,7";"5,8"
\ar@[green]@/_/@{->} "5,8";"6,9"
%now down and across
\ar@[green]@/^/@{->} "6,9";"6,10"
\ar@[green]@/_/@{->} "6,10";"7,10"
%blue path
\ar@[blue]@/^/@{->} "2,2";"3,3"
\ar@[blue]@/^/@{->} "3,3";"4,3"
\ar@[blue]@/^/@{->} "4,3";"5,4"
\ar@[blue]@/^/@{->} "5,4";"5,5"
\ar@[blue]@/^/@{->} "5,5";"6,6"
\ar@[blue]@/^/@{->} "6,6";"7,7"
\ar@[blue]@/^/@{->} "7,7";"7,8"
\ar@[blue]@/^/@{->} "7,8";"7,9"
\ar@[blue]@/^/@{->} "7,9";"7,10"
%extra blue paths
\ar@[blue]@/^/@{->} "3,3";"4,4"
\ar@[blue]@/^/@{->} "4,4";"5,4"
%next extra blue
\ar@[blue]@/^/@{->} "6,6";"6,7"
\ar@[blue]@/^/@{->} "6,7";"7,8"
%orange path
\ar@[orange]@/_/@{->} "2,2";"3,3"
\ar@[orange]@/_/@{->} "3,3";"4,3"
\ar@[orange]@/_/@{->} "4,3";"5,4"
\ar@[orange]@/_/@{->} "5,4";"5,5"
\ar@[orange]@/_/@{->} "5,5";"6,6"
\ar@[orange]@/_/@{->} "6,6";"6,7"
\ar@[orange]@/_/@{->} "6,7";"7,8"
\ar@[orange]@/_/@{->} "7,8";"7,9"
\ar@[orange]@/_/@{->} "7,9";"7,10"
\end{xy}
[/ilmath]
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