# Diseases and blue marks

Score Points: 5Solvers: 12

45 Natives in a certain colony have a myth that there is a certain disease signalled by a blue mark on their forehead. All the natives have blue marks. If they find out that they have the disease, they must declare it the same day in the town hall meeting, and leave the colony. However, no one else tells others if they have a blue mark out of courtesy. One day, a foreigner comes in, and tells the whole society that he has seen at least one blue mark in the colony. If the day the foreigner came was day 1, on what day will the first person leave the village? If you think they will never leave, enter 0.

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Work with smaller numbers first. What if there were only 2 natives? 3 natives?

Supposing there were just 2 natives, both would see one person with a blue mark, so each of them would realize that the two possibilities are –

This is from the point of view of 2. Now on the first day, none of them leave. 2 realises that if only 1 had a mark, on the first day, he would have
1.Seen no marks
2.Known at least one person has a mark
3.Hence concluded he has a mark and left
Since 1 didn’t leave, 2 must also have a mark. 1’s reasoning will be similar and both will leave on day 2.
Similarly if there were 3 people, person 3 would see 2 marks.
He would assume the possibilities were

1 |
2 |
3 |

Blue |
Blue |
Fine |

Blue |
Blue |
Blue |

3 will reason as follows: IN the case that 3 is fine,
1.If 1 and 2 saw only one mark each, they would do nothing on the first day.
2.On the second day, both of them would realise that since no one left on the first day, there must be 2 marks, and both would leave.
Both didn’t leave on day 2, which signifies that each of them did not see just 1 mark, which signifies that he must have a mark. He will leave on day 3.
All three of them will reason similarly and all of them will leave on day 3. The above logic can be extrapolated for 45 natives. All of them will leave on day 45.

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