P1118 [USACO06FEB] Backward Digit Sums G/S

`FJ` and his cows enjoy playing a mental game. They write down the numbers from $1$ to$ N(1 \le N \le 10)$ in a certain order and then sum adjacent numbers to produce a new list with one fewer number. They repeat this until only a single number is left. For example, one instance of the game (when $N=4$) might go like this: ```cpp 3 1 2 4 4 3 6 7 9 16 ``` Behind `FJ`'s back, the cows have started playing a more difficult game, in which they try to determine the starting sequence from only the final total and the number $N$. Unfortunately, the game is a bit above `FJ`'s mental arithmetic capabilities. Write a program to help `FJ` play the game and keep up with the cows.

Line 1: Two space-separated integers: N and the final sum.

Line 1: An ordering of the integers 1..N that leads to the given sum. If there are multiple solutions, choose the one that is lexicographically least, i.e., that puts smaller numbers first.

- For $40\%$ of the data, $1\le n\le 7$; - For $80\%$ of the data, $1\le n \le 10$; - For $100\%$ of the data, $1\le n \le 12$, $1\le sum\le 12345$.