Random Acts of Kindness

Question

Suppose one person decides to carry out a random act of kindness for three people on Monday.  On Tuesday, that same person is kind to three more people, and the three he/she helped on Monday each help three more individuals.  If the same circumstances were to occur on Wednesday, how many random acts of kindness would have been distributed?

Answer 1

Let's start with Person A. We will number each new person Pnn by the act of kindness performed on them. The notation Pxx-Pyy means that Personxx performs an act of kindness on Personyy.

Monday [“for three people on Monday”]
  A-P1
  A-P2
  A-P3

Tuesday [“same person is kind to three more people”]

  A-P4
  A-P5

  A-P6

        [“three he/she helped on Monday each help three more:]

  P1-P7

  P1-P8

  P1-P9

  P2-P10

  P2-P11

  P2-P12

  P3-P13

  P3-P14

  P3-P15

       Wednesday [:same circumstances were to occur on Wednesday”]

  A-P16

  A-P17

  A-P18

  P1-P19

  P1-P20

  P1-P21

  P2-P22

  P2-P23

  P2-P24

  P3-P25

  P3-P26

  P3-P27

  P4-P28

  P4-P29

  P4-P30

  P5-P31

  P5-P32

  P5-P33

  P6-P34

  P6-P35

  P6-P36

  P7-P37

  P7-P38

  P7-P39

  P8-P40

  P8-P41

  P8-P42

  P9-P43

  P9-P44

  P9-P45

  P10-P46

  P10-P47

  P10-P48

  P11-P49

  P11-P50

  P11-P51

  P12-P52

  P12-P53

  P12-P54

  P13-P55

  P13-P56

  P13-P57

  P14-P58

  P14-P59

  P14-P60

  P15-P61

  P15-P62

  P15-P63

Since the acts of kindness propagate, it is easier to compute the number of direct acts of kindness for each individual:
           Mon Tue Wed
Initial    3     3     3
M1               3     3
M2               3     3
M3               3     3
T01                     3
T02                     3
T03                     3
T04                     3
T05                     3
T06                     3
T07                     3
T08                     3
T09                     3
T10                     3
T11                     3
T12                     3

Total = 3 + 4*3 + 16*3 = 3+12+48 = 63

Answer 2

On Monday, 3 people were affected by random acts of kindness by a person - lets call him Bob. On the next day Bob affected 3 more people, for a total of 6 people affected. Then it says out of the first 3 people Bob affected on Monday, each of them affected three others. This means a total of 15 people were affected on Tuesday (6+3+3+3=15). On Wednesday, similar circumstances occur. So on Wednesday, Bob affects 3 more people, for a total of 18 people. Each of the Tuesday people who were affected each affect 3 more, for a total of 27 (3 people on Tuesday affect 3 each is 9, so 18+9=27). However, we have to include the people who were affected on Monday, like in the previous situation. There were 12 people here, who each affect 3 for 36 more people. Hence, 27+36=63 people affected total. This problem is much easier to visualize with a tree diagram. Just branch out the number of acts of kindness on each day for a visual representation and count them up.