# Are you smarter than a 8th grader?

**Logic**Level 5

\(1. X(1) = 1 \)

\(2.\) We define \(X(n+1)\) as a natural number, which can be obtained by replacing the digits of \(X(n)\) with \(1\) if the digit is \(0\), and with \(10\) if the digit is \(1\).

For example, \(X(1)=1\), \(X(2)=10\), \(X(3)=101\), \(X(4)=10110\) and so on.

\(A(n)\) is defined as the number of digits in \(X(n)\).

\(B(n)\) is defined as the number of times \(01\) appears in \(X(n)\).

For example, \(B(1)=0\), \(B(2)=0\), \(B(3)=1\), \(B(4)=1\), \(B(5)=3\) and so on.

What is \(A(23)+B(23)\)?

###### Note: There have been slight edits to the original question. The original question asks for a general formula of \(A(n)\) and \(B(n)\). Therefore, to provide for a numerical answer, the edits are necessary.

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###### This algebra problem appears in the anime *Puella Magi Madoka Magica*. Students attending the Math class are just 14 years old, yet they are expected to solve the following question on the spot!

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###### Translation credits: Puella Magi Wiki.

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