Handshakes

A Maths Starter of The Day

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

 

 

 

 

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Keywords: Starter | Investigations | Number Patterns | Problem Solving

  • Mr Martin Magic Maths Set, Dulwich College Beijing
  • Tuesday, November 23, 2010
  • "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
  • Monday, November 14, 2011
  • "
    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."

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Answers

If there was only one student in the room there would be no one to shake hands with.

If there were only two students in the room, there would be one handshake.

If there were three students, let's call them Ann, Bee and Cindy then:

Ann would shake hands with Bee
Ann would shake hands with Cindy
Bee would shake hands with Cindy

...so there would be three handshakes.

Here's a table showing these results:

No. of Students

 

No. of Handshakes

. . .

1

 

0

2

 

1

3

 

3

4

 

6

5

 

10

6

 

15

7

 

21

8

 

28

9

 

36

10

 

45

11

 

55

12

 

66

13

 

78

14

 

91

15

 

105

16

 

120

17

 

136

18

 

153

19

 

171

20

 

190

21

 

210

22

 

231

23

 

253

24

 

276

25

 

300

26

 

325

27

 

351

28

 

378

29

 

406

30

 

435

31

 

465

32

 

496

Can you find a formula?

If instead of handshakes, each student in the room gave every other student a present, how many presents would be given?