### The Lady or the Lions

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path to follow. Which room should she wait in to give her lover the greatest chance of finding her?

### Master Minding

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

### Two's Company

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

# Flippin' Discs

##### Stage: 3 Challenge Level:

Tamsin from Ringmer Primary School correctly states that

With 2 discs you win approximately half the time.

Congratulations to Thomas from Dalton School in New York for his correct solution:

The probability of obtaining the same colour with two discs is 1/2 because there are two ways of obtaining this result (red-red, green-green) and there are four combinations in total (red-red, green-green, red-green, and green-red). Hence, the probability is 2/4, which simplifies to 1/2.

For three discs, the probability is 1/4 using the same logic - there are two ways of obtaining the same colour and eight combinations in total (2 $\times$2 $\times$2, or 2$^3$).

For four discs, the probability is 1/8 because there are two ways of obtaining the same colour and sixteen combinations in total (2 $\times$2 $\times$2 $\times$2, or 2$^4$).

The general formula for the probability of flipping n discs and obtaining the same colour is:

$2/ 2^n$ or $1/ 2^{(n-1)}$.

Emma from St. Paul's Girls' School also found that

The probability of winning for n discs = 1/(2 to the power n-1)

Adan from Bancroft's uses Pascal's Triangle to help him determine the total number of possible combinations:

Using Pascal's Triangle you can work out the chances of them being all the same.

Firstly you must match the number of discs with the line of that number but one higher.

For example, let's use 3 discs:
On the fourth line of the triangle it says "1 3 3 1".
You must then add them all together, (that makes 8) this is the bottom of your fraction.
The left of the line is the number of combinations possible for them all being red.
The right is the number of combinations possible for them all being green.
The middle ones are the number of combinations possible for there being one of one colour and two of the other.
When you put one on the top and one on the bottom of your fraction you get the chances of that combination (eg.: the chances of there being one red and two green is 3/8).