## Solution to Puzzle #98: How Many Days to be Disease Free

Thanks again to Pallav for the puzzle – as I mentioned earlier, one of my favorites and a wonderful concept of mathematical induction to be taught to children. Only two people sent correct answers – Mohit Khare and Pratik Poddar – well done!

The answer is 23 days. For such puzzles, one can try a smaller problem first – i.e. what happens if there is only one person will yellow eyes, then for 2 and so on. Here is how it works (copying answer from Pratik Poddar directly):

Everyone knows that there is at least one diseased guy. Lets say that there was 1 diseased guy. That guy would see that there is no one else, and move out on day 1. If there were two diseased guys, no one would move out on day 1, as both think that there is someone else who is diseased. So, on day 2, both realise that both are diseased. Similarly, till day 22, all diseased people would be thinking that there are 22 diseased, and all will leave on day 22. But when no one leaves, all 23 diseased people leave on day 23.

Hope you enjoyed the puzzle!

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### 3 Responses to Solution to Puzzle #98: How Many Days to be Disease Free

1. Bharat says:

Why only 23 people will leave ?All 100 people should move out since no one knows the color of their own eyes.So the thought process should be same for everyone.

• Alok Goyal says:

Bharat, the easiest way to think about this is when there is only one person with yellow eyes. 99 people see someone with a yellow eye. The one who has yellow eyes sees no one else with yellow eyes, knows that there has to be at least one person with yellow eyes and will therefore realize that he/she is the only person with yellow eyes and will therefore be the only person to leave. Hope this explains.

• Bharat says:

Thank you.I was missing the point that others will be observing 23 people with yellow eyes and hence,wouldn’t doubt themselves.