## We Got A Problem #11: Rock Rock Paper Scissors

Most people know Rock-Paper-Scissors (or its more fun cousin, Ninja-Cowboy-Bear).  Kate and Ogden are locked in a long battle of online Rock-Paper-Scissors, with a winner scoring 1 point, and a tie scoring nothing for either player.  Game’s to 100.

Ogden won the first series 100-77 by constantly looking for patterns in Kate’s choices. For the second series, Kate tries something interesting: she has no pattern.  She randomly selects rock, paper, or scissors with equal probability.  She even tells Ogden that she is doing this, and he can do nothing about it but sigh and accept that he cannot beat this “strategy”.  Kate wins the second series 100-95 by luck.

For the third series, Ogden changes a rule: if rock beats scissors, the winner scores 2 points instead of 1.  Other victories (scissors over paper, paper over rock) are still worth 1.

Now Kate’s panicked: if she still randomly selects, is she beatable in the long run, or not?  Can she alter the probabilities to make a new “strategy” that Ogden can’t beat?  It seems like it would be bad to play scissors so often, and she should probably play more rocks… or not?

Let us know whether you find any ways that Kate can play to force Ogden into equilibrium.

## We Got A Problem #10: Cliff Hangers!

My wife was recently playing the Cliff Hangers slot machine, based on a great game from The Price Is Right.  Here’s how the bonus game works:

The goal is to climb exactly to the 25th and final step on the mountain (while yodeling, of course).  You keep taking turns, picking from 5 doors, one each with the numbers 3, 5, 10, 15, and 20 on them.  Whatever pick you make, that’s the number of steps taken on that turn.  The doors are randomized, so you don’t know what you’re going to get, and you can get the same number more than once.

If after any number of turns your total is exactly 25 steps, you win.  If you go over 25, you lose.

There are clearly some ways to win the game with lucky picking, such as 5 + 20 or 10 + 10 + 5.  My wife won even though her first pick was a 3.

What is the probability of hitting 25 and winning this game?  Which of these three goals gives the greatest chance of winning: 18, 55, or 58?  If there were no stopping point, what is the long-term probability of hitting a high number like 507?

We’ll provide a solution next week that involves a single mathematical calculation.  It’s pretty awesome.