On The Octogram Of Seth LaPod

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The latest wager that the Baron put to Sir R----- had them competing to first chalk a triangle between three of eight coins, with Sir R----- having the prize if neither of them managed to do so. I immediately recognised this as the game known as Clique and consequently that Sir R-----'s chances could be reckoned by applying the pigeonhole principle and the tactic of strategy stealing. Indeed, I said as much to the Baron but I got the distinct impression that he wasn't really listening.

Figure 1: A Triangle Of Endings
Recall that the pigeonhole principle asserts that if \(n\) items are to be shared between \(m\) places then, if \(n\) is greater than \(m\), at least one place must have more than one of them. In this wager a line might be chalked from each coin to any of the seven others either in blue or in red and if we should put each line into one of a pair of sets associated with the blue and the red chalk then it must be the case that one of them should contain at least four lines.
Now let us take three lines from one of those sets and think upon whether or not they might be part of a triangle. If there were a line of the same colour between any of the coins that those lines were drawn to then we should trivially have one and, if not, then we might chalk lines between the three of them to have a triangle of the other colour, as shown by figure 1.
We may therefore conclude that the game could not have ended without either The Baron or Sir R----- having fashioned a triangle.

Finally, if Sir R----- should have had a winning strategy then the Baron could have stolen it for himself to ensure victory. Specifically he could have chalked any line for his first move and thereafter responded to Sir R-----'s moves according to his plan. If it never involved the Baron's first line then it must have been the case that he should have won by it. Contrarily, if it had done, then he could have randomly chosen a pair of coins to connect and consequently played by Sir R-----'s scheme once again to the same end.

It should consequently not have been possible for Sir R----- to first chalk a triangle of coins, or to have prevented the Baron from doing so, and I should have most emphatically urged him to decline the Baron's wager!
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