If all the students in this room shook hands with each other, how many handshakes would there be altogether?

## A Mathematics Lesson Starter Of The Day

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• Mr Martin Magic Maths Set, Dulwich College Beijing
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• My Y5 class found this very challenging but it really got them thinking. In the end we found the solution!
• Mr J Saye, Castle Rock High School
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• Katie S in my Year 8 Set 1 class found that if you do the multiplication for one less than how many there are in the group and half it you get the correct answer,
eg. For a class of 20 19 x 20 = 380 / 2 = 190
this is becasue the multiplication shows everyone shaking others' hands and them shaking back which would be a double hand shake which is why you half the answer.
• Mr Paul Bridges, Cantell Maths And Computing Cilloge
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• The formula for this is 0.5n^2 - 0.5n, where n is the number of people.
• Miss. Loi, 10Y3
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• We have a class of 30 here at Woodbridge High School in Essex.
Gintaras found that 30 x 29 then halved = 435
Therefore there would be 435 handshakes!!
We loved this starter....
• 7T3, Homewood School And Six Form Center
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• In our class we have 26 people and we times 26X26 and we got 325.
• Y6, Jubilee Primary School
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• We enjoyed this problem and came up with a similar method as mentioned above, but took a while to figure out the halving. One pupil also mentioned the mathematician Johann Carl Friedrich Gauss and his theory of rings (I have no idea what he was talking about, but he was right) Basically in our class we have 25 pupils present so he did 12 x 25 becahe had halved 24 (the number of one person's handshakes) then multiplied by the number in the class. He found 12 by doing 24+1, 23+2, 22+3 etc... until the two numbers touched.
• Miss Kaplan, Harrold Priory
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• Used this with my Y5 class during form time. It was great for eliciting maths discussions and getting them to really think about the problem. They needed a prompt to think about how many handshakes each person should do but we got there in the end! Same formula as everybody else.
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• Off-sp, Reddit
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• In a room filled with people, every one shakes hand with everybody else. In total there are 66 handshakes, so how many people are there in the room?

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## Extension 1

Can you find a formula for the number of handshakes given the number of students?

## Extension 2

Edward and Lucy invite four couples to dinner. Each person shakes hands only with the people that he or she has not met before. Edward then asks his wife and eight guests how many hands they shook, and he receives nine different answers. How many people did Lucy shake hands with?

Bellos, Alex. Can You Solve My Problems?: a Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles. Guardian Books, 2017.

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