Math Detours · Grade 5 · Around the corner · 12

Everyone shakes hands

Ten people, each shakes hands with every other. How many handshakes? Count it badly, then count it well.

Name ____________________Date ____________

Tonight's idea

Five friends meet. Each one shakes hands once with every other. How many handshakes is that?

ABCDE
One dot for each friend, one line for each handshake. The four orange lines are A's.

Count it badly. Each of the 5 friends shakes 4 hands, and \(5 \times 4 = 20\). But the picture has only 10 lines.

The bad count looked at every handshake from both ends. A shaking with B and B shaking with A are the same handshake. Everything was counted twice, so halve it: \(20 \div 2 = 10\).

Handshakes = people × (one fewer than the people) ÷ 2.

Count it well. Go round in order. A shakes 4 hands. B has already met A, so B has 3 new ones. C has 2, D has 1, and E has none left. \(4 + 3 + 2 + 1 = 10\).

Two honest counts of the same thing must agree. When they do, you can trust the answer.

Watch one: ten people

Each of the 10 people shakes 9 hands: \(10 \times 9 = 90\). Every handshake was counted twice, so there are \(90 \div 2 = 45\).

Check it the other way. The first person shakes 9 hands, the next has 8 new ones, and so on down to 1.

\[9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45\]

So handshakes give a fast way to add up a run of numbers. There is more on that in a grade 6 sheet.

Your turn

  1. (a) Draw 4 dots and join every pair. How many handshakes for 4 people? (b) Do the same for 6 people. Check both with the rule.
  2. There are 20 children in a class. Each one shakes hands with every other. How many handshakes?
  3. (a) Eight teams are in a league. Each team plays every other team once. How many games? (b) At a party everybody shook hands with everybody else. There were 21 handshakes. How many people were there?
  4. (a) Six people are in a room. Each one shakes hands with exactly 3 of the others. How many handshakes? (b) Five people each say, "I shook hands with exactly 3 others." Explain why somebody must be wrong.

Stretch: diagonals

Draw a shape with 10 corners and 10 straight sides. A diagonal joins two corners that are not next to each other. How many diagonals does the shape have? Use handshakes, not a ruler.

Answers

  1. (a) 6. Check: \(4 \times 3 \div 2 = 6\), or \(3 + 2 + 1 = 6\). (b) 15. Check: \(6 \times 5 \div 2 = 15\), or \(5 + 4 + 3 + 2 + 1 = 15\).
  2. \(20 \times 19 = 380\), and half of that is 190.
  3. (a) A game is like a handshake between two teams: \(8 \times 7 \div 2 = 28\) games. (b) Try some numbers. 6 people give 15 and 7 people give \(7 \times 6 \div 2 = 21\). There were 7 people.
  4. (a) \(6 \times 3 = 18\) counts every handshake twice, so there were 9. (b) \(5 \times 3 = 15\) would count every handshake twice. But 15 is odd, so it cannot be twice a whole number of handshakes.

Stretch. Joining every pair of the 10 corners gives \(10 \times 9 \div 2 = 45\) lines. Ten of those are the sides of the shape. The rest are diagonals: \(45 - 10 = 35\).

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