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Maths Puzzle

  • 08-07-2020 7:55pm
    #1
    Registered Users Posts: 373 ✭✭


    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).


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