## Solution to Puzzle #95: How Many Ways to Play the Tabla

Thanks for a record breaking viewership this week…this is the maximum traffic I have had on the site ever since I started the puzzle! Many people sent the correct answer (which is 34 ways to do it). Some people gave a very elegant answer to the puzzle which include Amit Jain, Mohit Khare, Pratik Poddar and Vijay Venkat Raghavan – hats off to all of you! This puzzle was the origination of Fibonacci Series (en.wikipedia.org/wiki/Fibonacci_number) which is named after Fibonacci and documented this in 1202. However, this was originally found with music as its origin in 200 BC, but more formally articulated by Hemachandra in 1150 A.D. A link to the video from famous mathematician Manjul Bhargava, who in his interview with NDTV talked about it. https://www.youtube.com/watch?v=oCpdAPt7UOs Before describing the method with Fibonacci series, I want to outline the method that most people used. The correct answers here included Anirudh Baddepudi, Karan Sharma, Amit Mittal and Pallav Pandey. The logic to arrive at the answer (Using Mohit Khare’s answer): Possible combinations of A and B are in {8,0} , {6,1} , {4,2}, {2,3} and {0,4}.  Corresponding to each , following are the number of ways to arrange As and Bs: {8,0} := 1 [ chose 8 from 8 ] {6,1} := 7 [ chose 6 from 7 ] {4,2} := 15 [ chose 4 from 6 ] {2,3} := 10 [ chose 2 from 5 ] {0,4} := 1 [ chose 0 from 4 ] So total is : 34 The more elegant way to solve this is using Fibonacci Series. Lets call F(8) the number of ways to solve the problem. If we think about it, F(8) can be derived from the sum of following: – All the ways in which F(6) can be done with a B appended at the end of it, and – All the ways in which F(7) can be done with an A appended at the end of it. Therefore F(8) = F(7) + F(6), which is essentially the Fibonacci series. We know that F(1) = 1 (only one A) and F(2) = 2 (either two As or 1 B). The series will look as follows: 1, 2, 3, 5, 8, 13, 21, 34…. Hope you all enjoyed the puzzle.

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