# Three players were presented with 3 red balls and 2 yellow balls, 3 balls were chosen at random

and each was concealed in a different box. The challenge was for each player in turn to look inside two of the boxes to see if they could determine the color of the ball in the other box.

The first player looked insides boxes 1 and 3, but was not able to determine the color of the ball in box 2. The second player, having watched player one, looked inside boxes 2 and 3, but could not determine the color of the ball in box 1. Having watched both of the players one and two, the third player stated the color of the ball in one of the boxes, without bothering to look inside the boxes. Which box did she name and what was the color of the box? For player 1, box 1 and 3 can never be yellow at the same time, for if they were then they could have worked out box 2.

$$\begin{array}{c|lcr} box & \text{1} & \text{2} & \text{3} \\ \hline & R & & R \\ & R & & Y \\ & Y & & R \end{array}$$

For player 2, box 2 and 3 can never both be yellow otherwise they could have worked out box 1. So the possibilities are

$$\begin{array}{c|lcr} box & \text{1} & \text{2} & \text{3} \\ \hline & & R & R \\ & & R & Y \\ & & Y & R \end{array}$$

Now when I combine the info I get $$\begin{array}{c|lcr} box & \text{1} & \text{2} & \text{3} \\ \hline & R & R & R \\ & R & R & Y \\ & Y & Y & R \end{array}$$

Solution:\since the first player could not determine the color of the ball in box 2, boxes 1 and 3 contained R/Y.Y/R, or R/R (if they had contained Y/Y, he would have known box 2 contained a red ball).

If box 3 had contained a yellow ball, the second player would have been able to determine that box 1 contained a red ball. Since he could not do so, box 3 must contained a red ball.

This answer is succinct and I get it, so I want to know where I need to improve in my reasoning. I should be able to eliminate the R R Y row but I was stuck on it all night. I need to solve problems like this in under 1 min so any ideas/thoughts help. It still hasn't fully clicked yet, so it will be interesting to see different ways of explaining this concept.

## 1 Answer

First off all, you made a mistake when you combined the info. You have for player 1 and 2 \begin{align} \begin{array}{c|lcr} \text{box} & \text{1} & \text{2} & \text{3} \\ \hline & R & & R \\ & R & & Y \\ & Y & & R \end{array}& & \begin{array}{c|lcr} \text{box} & \text{1} & \text{2} & \text{3} \\ \hline & & R & R \\ & & R & Y \\ & & Y & R \end{array} \end{align} Now, the notation suggest that when you combine this, there are 3 possible combinations. However, we can mix the possibilities of the red ball in box 3 to obtain 5 different combinations \begin{array}{c|lcr} \text{box} & \text{1} & \text{2} & \text{3} \\ \hline & R & R & R \\ & Y & R & R \\ \hline & R & Y & R \\ & Y & Y & R \\ \hline & Y & R & Y \end{array} So, if player 2 had two red balls, player 1 could have a red or yellow ball in box 1, so player 2 doesn't know what is in box 1. If player 2 had a yellow ball in box 2 and a red ball in box 3, player 2 again doesn't know what is in box 1.
If player 2 has a red ball in box 2 and a yellow ball in box 3, player 1 must have had a red ball in box 1, so player 2 would know what is in box 1.

As player 2 doesn't know what is in box 2, there must be a red ball in box 3.

• excellent answer! – anna_xox Jul 11 '16 at 22:58