This problem was an instance of Post
correspondence problem. You have to find such sequence a[1], a[2], ...,
a[n], that strings x_{a[1]}x_{a[2]}...x_{a[n]}
and y_{a[1]}y_{a[2]}...y_{a[n]} are equal. There doesn't
exist an algorithm for this problem in general.

Let the sequence from the problem statement be x_{1},x_{2},...,x_{334}
and let the sequence from the wrong program be
y_{1},y_{2},...,y_{334}. Our first observation is that a[1]
must be 333, because we can't begin with nothing else. Also a[n] has to be 334.
For i less than 333 it is true that the size of x_{i} is equal
y_{i} (in fact, the size of strings x_{333} and y_{334} is 1
and the size of all other strings is 16). So after we add x_{i},
y_{i}, i<333 to both strings, the difference of their lengths won't
change.

Our algorithm will be a breadth-first-search where vertices of graph will be
strings. Let the vertex be a string **r**. That means
x_{a[1]}...x_{a[i]}=y_{a[1]}...y_{a[i]}r for
some a[1], ..., a[i]. When we add a[i+1] to the sequence, we get new string **g**:
x_{a[1]}...x_{a[i]}x_{a[i+1]} =
y_{a[1]}...y_{a[i]}y_{a[i+1]}g. Our goal is to find the
shortest sequence a[1], ..., a[n] such that the corresponding string is empty
− x_{a[1]}...x_{a[i]} = y_{a[1]}...
y_{a[n]} and the sequence a[1],...,a[n] is the right input for both
programs.

All the strings we get during the breadth-first-search will be of length 15 or
zero. We have to start with x_{333} and y_{333}. The difference
of their lengths is 16− 1=15. We can continue with x_{i},
y_{i}, i< 333, their difference of lengths is 0. We can also continue
with string x_{334} and y_{334}, their difference of lengths is
−15. We can't add number 333 to the sequence when the difference of
lengths is 15, because there will be string "22" in the sequence and there will
be no "22" in the corresponding part of x_{a[1]}...x_{a[i]}. So
no string during the breadth-first-search will be of size 30 and more.

Our breadth-first-search will find the shortest way to empty string. We only have to reconstruct the sequence a[1], ..., a[n] and output it.

We have to find out, what does the program do. First it checks whether
the input has 50≤ N ≤ 100. Then it transforms the original bipartite graph
into normal graph and checkes whether the graph has at least
N^{2}+2N edges.

If all these tests succeed, it runs a modified Depth-first-search. If you take a better look at it you can recognize code for finding bridges in a graph. The number of bridges in the graph is stored in variable cnt. If the number of bridges is at least 2N, the programs outputs it.

So we have to find a bipartite graph with at least
**N**^{2}+2**N** edges and at least 2N bridges. The maximum
matching can't be more than 2**N**, so the number of bridges has to be
exactly 2**N**. Here is a solution for **N**=4 (the idea can be used for
larger **N**):