Maths Puzzle

emanresu

I came across this puzzle in a book.

If everyone in a room shakes hands with every other person in the room, and there are 66 handshakes, how many people are in the room.

I can work out the answer logically by trial and error, and in a backwards way, but I would like to know the mathematical way to do it.

So if there are 3 people, I can call them A, B and C, and write down that A and B shake hands, A and C shake hands, and B and C shake hands, so there are 3 handshakes.
And I can continue like this with any number of people,
and eventually will come up with the answer.

2 people: A,B
AB = 1

3 people: A,B,C
AB AC = 2
BC = 1
Total = 3

4 people: A,B,C,D
AB AC AD = 3
BC BD = 2
Total = 5

5 people: A,B,C,D,E
AB AC AD AE = 4
BC BD BE = 3
CD CE = 2
DE = 1
Total = 10

6 people: A,B,C,D,E,F
AB AC AD AE AF = 5
BC BD BE BF = 4
CD CE CF = 3
DE DF = 2
EF = 1
Total = 15

7 people: A,B,C,D,E,F,G
AB AC AD AE AF AG = 6
BC BD BE BF BG = 5
CD CE CF CG = 4
DE DF DG = 3
EF EG = 2
FG = 1
Total = 21


12 people: A,B,C,D,E,F,G,H,J,K,L,M
AB AC AD AE AF AG AH AJ AK AL AM = 11
BC BD BE BF BG BH BJ BK AL AM = 10
CD CE CF CG CH CJ CK CL CM = 9
DE DF DG DH DJ DK DL DM = 8
EF EG EH EJ EK EL EM = 7
FG FH FJ FK FL FM = 6
GH GJ GK GL GM = 5
HJ HK HL HM = 4
JK JL JM = 3
KL KM = 2
LM = 1
Total =66

But I would like to know how to solve this mathematically by starting with the total number of handshakes (66) and finding the number of people in the room (12).

filbert the fox

Are you the only one who doesn't know that handshaking in the current crisis promotes the spread of Covid-19

namenotavailablE

I think (n^2-n)/2 will give the number of handshakes where n equals the number of people.

By the way, for 4 people, the OP missed C shakes the hand of D