# Gray Code

All of you know about Gray Code. It is a number code where consecutive numbers are represented by binary patterns that differ in one bit position only. In the following 4 examples of 3-bit gray code are shown: 000 000 000 000 001 001 010 010 011 011 011 011 010 010 001 001 110 110 101 101 111 100 100 111 101 101 110 110 100 111 111 100 In this problem we will deal with a gray code generation logic. This logic will generate the n-bit gray code using the coding of (n − 1) bits. Lets formally define the rules : • Each gray code has a starting bit pattern. Such as ‘0 0 0’ or ‘1 0 1’, etc. • An n-bit gray code will have 2n rows and two consecutive rows will differ by only one bit. • Each bit pattren will be present exactly once. • Gray code for 1-bit is trivial. Start with a bit and invert it in the next row. • To construct n-bit gray code keep any of the n bits fixed (either 0 or 1) for the first 2(n−1) rows and use (n − 1)-bit gray code (generated using this logic) for remaining (n − 1) bits. Then invert the fixed bit for the next 2(n−1) rows and also use (n − 1)-bit gray code for remaining (n − 1) bits whose bit pattern of the first row is the same as the bit pattern of the last row of previous 2(n−1) rows. For example 2-bit gray code starting with ‘00’ may be: 00 00 01 10 11 Or 11 10 01 Simmilarly 2-bit gray code starting with ‘01’ may be: 01 01 00 11 10 Or 10 11 00 If you observe carefully, you will see that the 3-bit gray codes given above are also constructed using this logic. Many such gray codes are possible for a particular starting bit pattern. We can order them from 1 to G(n) where G(n) denotes the number of such gray codes for n-bit. In our ordering scheme:

2/3 • 1st n-bit gray code has its leftmost bit fixed and it uses 1st (n − 1)-bit gray code for upper half and also 1st (n − 1)-bit gray code for lower half. • G(n − 1)-th n-bit gray code has its leftmost bit fixed and it uses 1st (n − 1)-bit gray code for upper half and G(n − 1)-th (n − 1)-bit gray code for lower half. • [G(n − 1) + 1]-th n-bit gray code has its leftmost bit fixed and it uses 2nd (n − 1)-bit gray code for upper half and 1st (n − 1)-bit gray code for lower half. • G(n)-th n-bit gray code has its rightmost bit fixed and it uses G(n − 1)-th (n − 1)-bit gray code for both halves. You have to find a n-bit gray code for given starting bit pattern and index. Input The first line of the input file contains a single integer N (0 < N ≤ 1000) which denotes the number of inputs. Each of the next N lines contains a string of bits for starting bit pattern and an integer for index. Number of bits will be between 1 to 6. And the index will be valid. Output Print the gray code for the given starting bit pattern and index. Put a blank line between two consec- utive sets of inputs. Sample Input 3 000 1 111 5 10 2 Sample Output 000 001 011 010 110 111 101 100 111 110 010 011 001 000 100 101 10 00

3/3 01 11