Regular octagon abcdefgh has its center at z . Each of the vertices and the center are to be associated with one of the digits 1 through 9, with each digit used once, in such a way that the sums of the numbers on the lines aze, bzf, czg, and djh are all equal. In how many ways can this be done?

brianlaw Mar 5, 2020

#1**+2 **

**Regular octagon abcdefgh has its center at z . Each of the vertices and the center are to be associated with one of the digits 1 through 9, with each digit used once, in such a way that the sums of the numbers **

**on the lines aze, bzf, czg, and dzh are all equal. In how many ways can this be done?**

\(\begin{array}{|r|c|c|c|c|c|c|c|c|c|} \hline \text{sum} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 1 & - & 3 & 4 & 5 & 6 & 7 & 8 & \color{red} 9 & \color{red}10 \\ \hline 2 & & - & 5 & 6 & 7 & 8 & \color{red}9 & \color{red}10 & \color{red}11 \\ \hline 3 & & & - & 7 & 8 & \color{red}9 & \color{red}10 & \color{red}11 & 12 \\ \hline 4 & & & & - &\color{red} 9 & \color{red}10 & \color{red}11 & 12 & 13 \\ \hline 5 & & & & & - & \color{red}11 & 12 & 13 & 14 \\ \hline 6 & & & & & & - & 13 & 14 & 15 \\ \hline 7 & & & & & & & - & 15 & 16 \\ \hline 8 & & & & & & & & - & 17 \\ \hline 9 & & & & & & & & & - \\ \hline \end{array} \)

We need equal sums in four different ways: There are three possible sums: **9**, **10**, and **11**.

\(\begin{array}{|rcll|} \hline \text{sum }~=9 \\ \hline 9 &=& 1+8 \\ 9 &=& 2+7 \\ 9 &=& 3+6 \\ 9 &=& 4+5 \\ \hline \text{There are }\\ ~ 4! = 24~ \text{ ways} \\ \hline \end{array} \begin{array}{|rcll|} \hline \text{sum }~=10 \\ \hline 10 &=& 1+9 \\ 10 &=& 2+8 \\ 10 &=& 3+7 \\ 10 &=& 4+6 \\ \hline \text{There are }\\ 4! = 24~ \text{ ways} \\ \hline \end{array} \begin{array}{|rcll|} \hline \text{sum }~=11 \\ \hline 11 &=& 2+9 \\ 11 &=& 3+8 \\ 11 &=& 4+7 \\ 11 &=& 5+6 \\ \hline \text{There are }\\ ~ 4! = 24~ \text{ ways} \\ \hline \end{array}\)

In total so far these are \(3\times 4!=3\times 24 = \mathbf{72} \text{ ways}\).

**Swapping the two elements of a pair of numbers:**

\(\begin{array}{|rcll|} \hline \text{One of the $\mathbf{72}$ ways} \\ \text{swapping the two elements of a pair of numbers} \\ \hline \mathbf{ab} & \mathbf{cd} & \mathbf{ef} & \mathbf{gh} \\ ab & cd & ef & hg \\ ab & cd & fe & gh \\ ab & cd & fe & hg \\ \hline ab & dc & ef & gh \\ ab & dc & ef & hg \\ ab & dc & fe & gh \\ ab & dc & fe & hg \\ \hline ba & cd & ef & gh \\ ba & cd & ef & hg \\ ba & cd & fe & gh \\ ba & cd & fe & hg \\ \hline ba & dc & ef & gh \\ ba & dc & ef & hg \\ ba & dc & fe & gh \\ ba & dc & fe & hg \\ \hline \text{There are }\\ ~ \mathbf{16}~ \text{ ways} \\ \hline \end{array}\)

This can be done in \(16\times 72 = \mathbf{1152}\) ways.

heureka Mar 5, 2020