# Looking at patterns on an Othello board and also the Birthday Paradox

For today’s Family Math project I put an Othello board down in front of the kids and asked them to come up with their own math question. It took a little bit of thinking on their end, but each kid came up with a really interesting question:

(1) Younger son: If you placed the tiles on the board and randomly, what is the probability that all of the tiles would have the white side facing up?

(2) Older son: If all of the tiles have white facing up, and select one tile at random 32 times in a row – what is the probability that at the end of this process you’ll have exactly 32 black tiles and 32 white tiles facing up?

Here’s that conversation:

We tackled my younger son’s problem first. The main problem solving ideas here were reducing the problem to a problem that was easier to solve and then a little bit of pattern recognition.

My older son’s problem was next. We spent a few minutes making sure that we were clear about the problem, and then began looking at the problem. I asked my older son for a guess at the probability and he came up with a very thoughtful explanation of why he expected the probability to be small. We then looked carefully at his explanation.

To wrap up, we went to Wolfram Alpha to calculate the number we’d found in the last video. We also spent a couple of minutes repeating the calculation for the Birthday Paradox problem – how many people do you need to have in a room in order to have a 50% chance that two people have the same birthday?

So, a really fun morning working through the questions the boys came up with. Amazing the fun you can have just looking at an Othello board.