Handshakes

If a group of people at a party are all introduced to each other, and everyone shakes hands with everyone else exactly once, how many handshakes have taken place?

If we imagine ten people, and the first person shakes hands with everyone else, that will total nine handshakes (since he does not shake hands with himself).

If the second person now shakes hands with everyone, except for himself and the first person, with whom he has already shaken hands, this will add eight handshakes to the total.

Continuing along the line, each person will then add seven, then six, then five and so on until the final tenth person, who will already have shaken hands with everyone else and will add zero handshakes to the total.

This means to work out how many handshakes altogether, we just need to add together the numbers:
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0

This sum might be straightforward, but it will take a little time, and especially if we are dealing with a larger group of people, it will help to have an easier way to do this. In fact, it is simple. All we need to do is pair up the people so that 0 is paired with 9, 8 is paired with 1 and so on, so that each pair contains a total of 9 handshakes, which is one less than the number of people. The number of pairs will be exactly half the number of people, in this case 5. Then we just multiply 5 by 9 to get 45 handshakes in total.

If the number of people is odd, you can still divide this number by two (it will include a fraction of a half) - because when you pair them up, the one person left over in the middle will have exactly half as many handshakes as the person with the most handshakes, and so counts as half a pair.

This method can be used to work out the number of handshakes for any size of group - by multiplying the number of people minus one, by half the number of people. So, for 50 people we would simply need to multiply 49 by 25, giving 1225 handshakes. Or, for the 90,000 people in Wembley Stadium, we would need to multiply 89,999 by 45,000, giving 4,049,955,000 handshakes! In fact, this technique makes adding up all of the numbers from 1 to any number quite an easy task.

This method was discovered by German mathematician Carl Friedrich Gauss, when he was a child at school and his teacher set the class a task to add up all the numbers from 1 to 100, which he thought would take a long time. In fact, Gauss answered almost immediately. What two numbers would Gauss need to multiply together to get the answer?