Ohio University 2017-2018 #1

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2017-09-14 01:30 CEST

Ohio University 2017-2018 #1

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2017-09-14 03:40 CEST
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Problem E
Hamiltonian Hypercube

Hypercube graphs are fascinatingly regular, hence you have devoted a lot of time studying the mathematics related to them. The vertices of a hypercube graph of dimension $n$ are all binary strings of length $n$, and two vertices are connected if they differ in a single position. There are many interesting relationships between hypercube graphs and error-correcting code.

One such relationship concerns the $n$-bit Gray Code, which is an ordering of the binary strings of length $n$, defined recursively as follows. The sequence of words in the $n$-bit code first consists of the words of the $(n-1)$-bit code, each prepended by a $0$, followed by the same words in reverse order, each prepended by a $1$. The $1$-bit Gray Code just consists of a $0$ and a $1$. For example the $3$-bit Gray Code is the following sequence:

\[ 000,001,011,010,110,111,101,100 \]

Now, the $n$-bit Gray Code forms a Hamiltonian path in the $n$-dimensional hypercube, i.e., a path that visits every vertex exactly once (see Figure 1).

\includegraphics[width=0.33\textwidth ]{example-fig}
Figure 1: The $3$-dimensional hypercube and the Hamiltonian path corresponding to the $3$-bit Gray Code.

You wonder how many vertices there are between the vertices $0^ n$ ($n$ zeros) and $1^ n$ ($n$ ones) on that path. Obviously it will be somewhere between $2^{n-1}-1$ and $2^ n-2$, since in general $0^ n$ is the first vertex, and $1^ n$ is somewhere in the second half of the path. After finding an elegant answer to this question you ask yourself whether you can generalise the answer by writing a program that can determine the number of vertices between two arbitrary vertices of the hypercube, in the path corresponding to the Gray Code.

Input

The input consists of a single line, containing:

  • one integer $n$ ($1 \leq n \leq 60$), the dimension of the hypercube

  • two binary strings $a$ and $b$, both of length $n$, where $a$ appears before $b$ in the $n$-bit Gray Code.

Output

Output the number of code words between $a$ and $b$ in the $n$-bit Gray Code.

Sample Input 1 Sample Output 1
3 001 111
3
Sample Input 2 Sample Output 2
3 110 100
2