Heard this awesome puzzle yesterday while watching a video of Manjul Bhargava with my children (thanks to my wife Pooja to introduce me to this one). For those who do not know, Manjul Bhargava is a Princeton Professor who got the Field Medal in August last year, given for Mathematics once in 4 years. In his interview with NDTV, he asked this puzzle to children.
Assume that there are only two possible ways to hit the tabla – lets call them A and B. A requires one time unit and B requires 2 time units. You have a total of 8 time units and need to use A and B to fill these 8 time units. How many different ways are there.
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We can create the equation A+2B=8. Then after that we can look at the different cases for B:
If B is one, A must be 6. So we then need to arrange the B in the 6+1=7 spaces. This gives us 7C1 possibilities If B is two, A must be 4. We then need to arrange the 2 B’s in 2+4=6 spaces. This gives us 6C2 possibilities.
Continuing this, we get 7C1+6C2+5C3+4C4 = 7+15+10+1=33 ways.
By construction, its Fibonacci sequence. f(8) is F(9)