Handshake+maths+(triangular+numbers)

Get the children into groups starting with 2 people, then3, then 4 etc. Tell them to each shake hands with every other person in their group ONCE. Get them to record how many handhsakes in total for the group. Then do the following worksheet/class discussion about the numbers they found for each group and the connection between them.



Sam is trying to work out how many handshakes there would be if 20 mathematicians met. He says that since each mathematician shakes hands 19 times, there must be   20   19   handshakes altogether. Helen disagrees; she worked out  19+18+17+  +2+1   and got a different answer. What is wrong with Sam's reasoning? How should he modify his method?

One day, 161 mathematicians met. How many handshakes took place this time?

Can you describe a quick way of working out the number of handshakes for any size of mathematical gathering?

Could there be exactly 4851 handshakes at a gathering where everyone shakes hands? How many mathematicians would there be?

What about the following numbers of handshakes?
 * 6214
 * 3655
 * 7626
 * 8656

You may wish to try the problems [|Picturing Triangle Numbers] and [|Mystic Rose]. Can you see why we chose to publish these three problems together?

You may also be interested in reading the article [|Clever Carl], the story of a young mathematician who came up with an efficient method for adding lots of consecutive numbers.