invalid data
0 = 1 * 0 + 0
1 = 1 * 1 + 0
invalid data
0 = 2 * 0 + 0
1 = 2 * 0 + 1
2 = 1 * 2 + 0
2 = 2 * 1 + 0
3 = 2 * 1 + 1
2 = 3 * 0 + 2