#### You may also like ### Stop or Dare

All you need for this game is a pack of cards. While you play the game, think about strategies that will increase your chances of winning. ### Snail Trails

This is a game for two players. You will need some small-square grid paper, a die and two felt-tip pens or highlighters. Players take turns to roll the die, then move that number of squares in a straight line. Move only vertically (up/down) or horizontally (across), never diagonally. You can cross over the other player's trails. You can trace over the top of the other player's trails. You can cross over a single trail of your own, but can never cross a pair of your trails (side-by-side) or trace over your own trail. To win, you must roll the exact number needed to finish in the target square. You can never pass through the target square. The game ends when a player ends his/her trail in the target square, OR when a player cannot move without breaking any of the rules. ### Odds and Evens

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

# Flippin' Discs

##### Age 11 to 14 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). 