Thread by @pgeerkens

@GWOMaths Hint follows

@GWOMaths Rochambeau is just a fancy name for Rock-Paper-Scissors.

@GWOMaths If you're having difficulty visualizing this,
consider the situation after Game 1 as the baseline.

@GWOMaths Then ask yourself:

1) How many ways are there of getting back to that baseline situation?

2) What occurs to the combined number of wins by Alana and Beth for each possibility?

@GWOMaths Solution follows

@GWOMaths After the first game:
Alana/Beth have accumulated one win against each other,
resulting in the winner staying in the arena
while the other retires to be replaced by Chen;
with one Alana v Beth game played.

AB_Games (G) = 1
AB_Wins (W) = 1

@GWOMaths Now Chen can win any number of games,
each of which has no affect on our two tracking stats above
but merely alternates which of Alana/Beth is in the arena and which is on the side-lines.

We still have
AB_Games (G) = 1
AB_Wins (W) = 1
throughout while Chen's win streak holds.

@GWOMaths Then Chen loses, and the following happens:
- AB_Wins increments to 2 (the win over Chen);
- Alana & Beth have another (their second) head-to-head match, so AB_Games increments to 2;
- AB_Wins increments to 3 (the win over the other);
- One of Alana/Beth retires on the loss.

@GWOMaths We are now back to an arena with Chen and either Alana or Beth, with our tracking stats at:

AB_Games (G) = 2
AB_Wins (W) = 3

This cycle repeats every time Chen loses, with
AB_Games incremented once and
AB_Wins incremented twice
(one over Chen and one for the head-to-head win).

@GWOMaths The pattern is that at all time the invariant
AB_Wins + 1 = 2 * AB_Games
holds:

After Game 1:
AB_Games (G) =1
AB_Wins (W) = 1

After Chen Loss #1
AB_Games (G) = 2
AB_Wins (W) = 3

After Chen Loss #2:
AB_Games (G) = 3
AB_Wins (W) = 5
:
:

@GWOMaths Our final situation is that AB_Wins = 20 + 23 = 43;
thus we get by applying our invariant
(viz AB_Wins + 1 = 2 * AB_Games)

AB_Games = (AB_Wins + 1) / 2
= 44 / 2
= 22.

Today's answer is that Alana and Beth played each other
22 times.

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