This problem looks simple, but for some reason I am stuck with it.
There is a circle of bits (0s and 1s) with the following constraints:
- There is no run of 4 or more consecutive identical bits.
- The number of $0$'s is $4$ plus the number of $1$'s.
Now we count the number of pairs of consecutive 0's and 1's. The pairs don't have to be disjoint (i.e. "000" counts as two pairs). My question is: can we prove that the number of $00$'s is larger then the number of $11$'s?