## Solution to Puzzle #174: Happy Birthday!

This puzzle was not an easy one, and one of the ways to make it easier was to solve it for a smaller number of candles and then try to solve for larger number of candles. There is a whole class of problems we can solve that way, and this method is called induction.

I received many correct answers and a few good attempts where the answers were not correct. Correct answers came from Abhinav Jain, Suman Saraf, Pratik Poddar and Praneeth. Very mentionable attempts  from Arushi & Ishir Gupta as well as Prakhar’s daughter. Well done all.

The answer is very close to 4! More precisely, it will take about 3.994987 blows. Why?

Let’s start with a smaller number of candles and work our way up. Suppose you have a cake with just a single candle. You’ll blow it out in one blow, for sure. Suppose there are two. Half the time you’ll blow them both out in one go, and half the time it’ll take two blows. Let’s make a list:

One candle: 1

Two candles: (1/2)⋅1+(1/2)⋅2=1.5
Three candles: (1/3)⋅1+(1/3)⋅(1+1.5)+(1/3)⋅(1+1)=1.83¯
Four candles: (1/4)⋅1+(1/4)⋅(1+1.83¯)+(1/4)⋅(1+1.5)+(1/4)⋅(1+1)=2.083¯

With each additional candle, you have an equal chance of blowing them out in one go and of only snuffing some specific number, leaving some to tackle on the next blow. Notice the pattern! For one candle, the average number of blows is one. For two, it’s 1+1/2. For three, it’s 1+1/2+1/3. For four, it’s 1+1/2+1/3+1/4. And so on. So to get the answer, we simply compute this harmonic sum:

∑  (1/i), i goes from 1 to 30 = ≈3.994987

Hope you all enjoyed the puzzle!

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