(related to Problem: The Cross Of Cards)
There are eighteen fundamental arrangements, as follows, where I only give the numbers in the horizontal bar, since the remainder must naturally fall into their places. $$\begin{array}{ccccc} 5&6& 1 &7& 4\\ 3&5& 1 &6& 8\\ 3&4& 1 &7& 8\\ 2&5& 1 &7& 8\\ 2&5& 3 &6& 8\\ 1&5& 3 &7& 8\\ 2&4& 3 &7& 8\\ 1&4& 5 &7& 8\\ 2&3& 5 &7& 8\\ 2&4& 5 &6& 8\\ 3&4& 5 &6& 7\\ 1&4& 7 &6& 8\\ 2 &3& 7 &6& 8\\ 2&4& 7 &5& 8\\ 3&4& 9 &5& 6\\ 2&4& 9 &5& 7\\ 1&4&9 &6& 7\\ 2 &3&9 &6&7\\ \end{array}$$ It will be noticed that there must always be an odd number in the center, that there are four ways each of adding up $23,$ $25,$ and $27,$ but only three ways each of summing to $24$ and $26.$
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