GeekDad Puzzle of the Week Solution – Roshambo Strategery

Last week’s puzzle, as presented:

Congratulations! You have just made the semi-finals for this year’s (un)Official GeekDad Paper-Scissors-Rock Tournament!

You are up against the same four opponents from last year — Inky, Blinky, Pinky, and Clyde. This is both good news, and bad news.

The good news is that not only do you know that they use a fixed strategy, but you happen what the strategy of each is from earlier play. Here is how each plays:

Inky always starts with Scissors, and then plays whatever you played last turn.
Blinky always starts with Paper, and then moves through Scissors and Rock in turn. When he completes the cycle, he repeats, ad infinitum.
Pinky always starts with Rock, and then plays what would beat what you played last turn.
Clyde always plays Paper. Always. (He’s not terribly bright, but somehow always gets this far in the tournament.)

The bad news (ok, maybe the interesting news?) is that you need to define a fixed strategy as well and implement that strategy against each of the the other four players in the tournament. To move forward into the next round, select a set of ten moves (either fixed, as in “PSRRSPPSSP”) or a definable algorithm (like those above) that gives you the most wins over the four players above.

You must select one common strategy against each of the four players, and should definitely show your work. That is, please don’t simply send in an algorithm or set of moves and leave it to us to determine how well it works against your opponents.

Reading through this past week’s submissions was a lot of fun; from efficient people submitting a set of throws that just barely beat two opponents (really straightforward, as Clyde always plays Paper) to solutions that optimized against all four opponents to actual strategies that tended not necessarily to optimize, but were interesting to “confirm.”

There are a total of 3¹⁰ or 59,049 possible “fixed-throw” solutions – there are 10 throws, and each has three independent choices. Reviewing and scoring each of these, there were exactly 200 solutions that beat all four competitors, 5,494 that beat three of the four, and 20,071 that beat two out of four. That’s right — if you simply threw a completely random hand, you had an almost 43% chance of beating two of your opponents.

Of the 200 solutions that beat all four opponents, the two most “winning” solutions (PSPSRPSRPS, RSPSRPSRPS) had 22 single-throw wins against the other players. Note that these solutions didn’t necessarily have the most single-throw wins — SRPSRPSRPS had 23 single-throw wins, but lost to Pinky.

Congratulations to Jordan Wirth, who not only submitted the SRPSRPSRPS solution, but also was randomly selected as the winner of this week’s $50 ThinkGeek Gift Certificate.

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