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On Triumvirate – student

student from thus spake a.k.

When last they met, the Baron invited Sir R----- to join him in a wager involving a sequence of coin tosses. At a cost of seven coins Sir R----- would receive one coin for every toss of the coin until a run of three heads or of three tails brought the game to its conclusion.

To evaluate its worth to Sir R----- we begin with his expected winnings after a single toss of the coin.
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Triumvirate – baron m.

baron m. from thus spake a.k.

Welcome Sir R-----! Pray join me for a draught of cider to refresh you on this close summer's eve!

Would you be in the mood for some sporting diversion?

It pleases me to hear so Sir! It pleases me greatly!

I challenge you to a game that reflects the somewhat unique political system adopted by the three sister-queens of Thornborough; Alnitak, Alnilam and Mintaka. Whilst ruling as a triumvirate their constitution requires all three to concur upon any decision, quite unlike any others in antiquity or modernity which, as I'm quite sure that you are aware, require but two.
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On The Octogram Of Seth LaPod – student

student from thus spake a.k.

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.
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The Octogram Of Seth LaPod – baron m.

baron m. from thus spake a.k.

Salutations Sir R-----! I trust that this fine summer weather has you thirsting for a flagon. And perhaps a wager?

Splendid! Come join me at my table!

I propose a game played as a religious observance by the parishioners of the United Reformed Eighth-day Adventist Church of Cthulhu, the eldritch octopus god that lies dead but dreaming in the drowned city of Hampton-on-Sea.
Several years ago, the Empress directed me to pose as a peasant and infiltrate their temple of Fhtagn in the sleepy village of Saint Reatham on the Hill when it was discovered that Bishop Derleth Miskatonic had been directing his congregation to purchase vast tracts of land in the Ukraine and gift them to the church in return for the promise of being spared when Cthulhu finally wakes and devours mankind.
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On Pennies From Heaven – student

student from thus spake a.k.

Recall that the Baron and Sir R-----'s most recent wager first had the Baron place three coins upon the table, choosing either heads or tails for each in turn, after which Sir R----- would follow suit. They then set to tossing coins until a run of three matched the Baron's or Sir R-----'s coins from left to right, with the Baron having three coins from Sir R----- if his made a match and Sir R----- having two from the Baron if his did.

When the Baron described the manner of play to me I immediately pointed out to him that it was Penney-Ante, which I recognised because one of my fellow students had recently employed it to enjoy a night at the tavern entirely at the expense of the rest of us! He was able to do so because it's an example of an intransitive wager in which the second player can always contrive to make a choice that will best the first player's.
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Pennies From Heaven – baron m.

baron m. from thus spake a.k.

Sir R-----, my good friend! Come shake the snow from your boots and join me by the hearth for a draught of warming spirits!

And will you also join me in a wager whilst you let the fire chase the chill from your bones?

Fine fellow! Stout fellow!

I have in mind a game that reminds me of my raid upon the vault of Heaven, which I mounted in order to make amends to the Empress for my failure to snatch the Amulet of Yendor from the inner circle of Hell.
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On Divisions – student

student from thus spake a.k.

The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.

Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.