Paths through the Hourglass

In the hourglass to the right a path is marked. A path always starts at the first row and ends at the last row. Each cell in the path (except the first) should be directly below to the left or right of the cell in the path in the previous row. The value of a path is the sum of the values in each cell in the path. A path is described with an integer repre- senting the starting point in the first row (the leftmost cell being 0) followed by a direction string containing the letters ‘L’ and ‘R’, telling whether to go to the left or right. For in- stance, the path to the picture is described as ‘2 RRRLLRRRLR’. Given the values of each cell in an hourglass as well as an integer S, calculate the number of distinct paths with value S. If at least one path exist, you should also print the path with the lowest starting point. If several such paths exist, select the one which has the lexicograph- ically smallest direction string. Input The input contains several cases. Each case starts with a line containing two integers N and S (2 ≤ N ≤ 20, 0 ≤ S < 500), the number of cells in the first row of the hourglass and the desired sum. Next follows 2N − 1 lines describing each row in the hourglass. Each line contains a space separated list of integers between 0 and 9 inclusive. The first of these lines will contain N integers, then N − 1, ..., 2, 1, 2, ..., N − 1, N. The input will terminate with N = S = 0. This case should not be processed. There will be less than 30 cases in the input. Output For each case, first output the number of distinct paths. If at least one path exist, output on the next line the description of the path mentioned above. If no path exist, output a blank line instead. Sample Input 6 41 672368 18071 2657 310 76 8

2/2 88 653 9595 64413 269438 27 31 2 35 5 26 28725 3602 134 25 3 72 293 1044 48723 00 Sample Output 1 2 RRRLLRRRLR 0 5 2 RLLRRRLR