## Sunday, September 4, 2016

### Count Pairs Of Consecutive Zeros - GeeksforGeeks

Count Pairs Of Consecutive Zeros - GeeksforGeeks
Consider a sequence that starts with a 1 on a machine. At each successive step, the machine simultaneously transforms each digit 0 into the sequence 10 and each digit 1 into the sequence 01.
After the first time step, the sequence 01 is obtained; after the second, the sequence 1001, after the third, the sequence 01101001 and so on.
How many pairs of consecutive zeros will appear in the sequence after n steps?

```Input : Number of steps = 3
Output: 1
// After 3rd step sequence will be  01101001

Input : Number of steps = 4
Output: 3
// After 4rd step sequence will be 1001011001101001

Input : Number of steps = 5
Output: 5
// After 3rd step sequence will be  01101001100101101001011001101001
```
This is a simple reasoning problem. If we see the sequence very carefully , then we will be able to find a pattern for given sequence. If n=1 sequence will be {01} so number of pairs of consecutive zeros are 0, If n = 2 sequence will be {1001} so number of pairs of consecutive zeros are 1, If n=3 sequence will be {01101001} so number of pairs of consecutive zeros are 1,
If n=4 sequence will be {1001011001101001} so number of pairs of consecutive zeros are 3.

So length of the sequence will always be a power of 2. We can see after length 12 sequence is repeating and in lengths of 12. And in a segment of length 12, there are total 2 pairs of consecutive zeros. Hence we can generalize the given pattern q = (2^n/12) and total pairs of consecutive zeros will be 2*q+1.
`int` `consecutiveZeroPairs(``int` `n)`
`{`
`    ``// Base cases`
`    ``if` `(n==1)`
`        ``return` `0;`
`    ``if` `(n==2 || n==3)`
`        ``return` `1;`

`    ``// Calculating how many times divisible by 12, i.e.,`
`    ``// count total number repeating segments of length 12`
`    ``int` `q = (``pow``(2,n) / 12);`

`    ``// number of consecutive Zero Pairs`
`    ``return` `2*q + 1;`
`}`