Problem B
Magic Bitstrings
A bitstring, whose length is one less than a prime, might be magic. 1001 is one such string. In order to see the magic in the string let us append a non-bit x to it, regard the new thingy as a cyclic string, and make this square matrix of bits
|
each bit |
1001 |
|
every 2nd bit |
0110 |
|
every 3rd bit |
0110 |
|
every 4th bit |
1001 |
This matrix has the same number of rows as the length of the original bitstring. The $m$-th row of the matrix has every $m$-th bit of the original string starting with the $m$-th bit. Because the enlarged thingy has prime length, the appended x never gets used.
If each row of the matrix is either the original bitstring or its complement, the original bitstring is magic.
Each line of input (except last) contains a prime number $p \leq 100000$. The last line contains 0 and this line should not be processed. For each prime number from the input produce one line of output containing the lexicographically smallest, non-constant magic bitstring of length $p-1$, if such a string exists, otherwise output Impossible.
| Sample Input 1 | Sample Output 1 |
|---|---|
5 3 17 47 2 79 0 |
0110 01 0010111001110100 0000100001101010001101100100111010100111101111 Impossible 001001100001011010000001001111001110101010100011000011011111101001011110011011 |
