Consider these 2 questions where recurrence relations can be applied:
Q1) Given an (nxm) where n denotes rows and m denotes columns of a grid, find the number of unique paths ($a_{n,m}$) that goes from the top left corner of the grid to the bottom right corner of the (nxm) grid. Rule: Can only move downwards or rightwards when travelling on the grid
For Q1) the solution is as follows:
- Base Cases: when n or m is equals to 1 (i.e. $a_{1,m} =1 $ and $a_{n,1} = 1$)
- Considering the last action: There are 2 cases - 1) moving downwards 2) moving rightwards to reach the bottom left corner of the grid. Hence, the recurrence relation is of the form: $a_{n,m} = a_{n-1,m} + a_{n,m-1}$.
But notice here that there is a 1-1 correspondence between $a_{n,m}$ and ($a_{n-1,m} + a_{n,m-1}$) -- for every unique path in $a_{n-1,m}$ we move 1 grid downwards and for every unique path in $a_{n,m-1}$ we move 1 grid rightwards - doing so will create a 1-1 correspondence with the unique paths in $a_{n,m}$.
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Q2) The "Tower of Hanoi" Problem where the recurrence relation is of the form: $a_n = 2a_{n-1} + 1$ where $a_n$ denotes the minimum number of steps needed to move n disks from one bar to another bar
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Although Q1) seems to have a 1-1 function underneath its recurrence relation, I am wondering whether the same can be argued for the "Tower of Hanoi" problem as I am not able to think of it..
