Celebrity Split

Jack and Jill have decided to separate and divide their property equally. Each of their $N$ mansions has a value between $1\, 000\, 000$ and $40\, 000\, 000$ dollars. Jack will receive some of the mansions; Jill will receive some of the mansions; the remaining mansions will be sold, and the proceeds split equally.

Neither Jack nor Jill can tolerate the other receiving property with higher total value. The sum of the values of the mansions Jack receives must be equal to the sum of the values of the mansions Jill receives. So long as the value that each receives is equal, Jack and Jill would like each to receive property of the highest possible value.

Given the values of $N$ mansions, compute the value of the mansions that must be sold so that the rest may be divided so as to satisfy Jack and Jill.

Suppose Jack and Jill own $5$ mansions valued at $6\, 000\, 000$, $30\, 000\, 000$, $3\, 000,000$, $11\, 000\, 000$, and $3\, 000\, 000$ dollars. To satisfy their requirements, Jack or Jill would receive the mansion worth $6\, 000\, 000$ and the other would receive both mansions worth $3\, 000\, 000$ dollars. The mansions worth $11\, 000\, 000$ and $30\, 000\, 000$ dollars would be sold, for a total of $41\, 000\, 000$ dollars. The answer is therefore $41\, 000\, 000$.

The input consists of a sequence of test cases. The first line of each test case contains a single integer $N$, the number of mansions, which will be no more than $24$. This line is followed by $N$ lines, each giving the value of a mansion and it will be at most $40\, 000\, 000$. The final line of input contains the integer zero. This line is not a test case and should not be processed. The sum of $N$ over all test cases is at most $50$.

For each test case, output a line containing a single integer, the value of the mansions that must be sold so that the rest may be divided so as to satisfy Jack and Jill.

Sample Input 1 | Sample Output 1 |
---|---|

5 6000000 30000000 3000000 11000000 3000000 0 |
41000000 |