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In previous "Crack the Combination" puzzles*, the combination has always been decimal, that is, the digits in the combination were the ten from 0 to 9. Now, the combination lock you need to open has a binary combination of five of the two binary digits―"bits"―0 and 1. The following are some incorrect combinations:
| Clue | Combination | # of bits correct |
|---|---|---|
| 1 | 1 0 0 0 0 | 1 |
| 2 | 1 0 0 1 0 | 10 |
| 3 | 0 0 1 1 1 | 11 |
| 4 | 1 0 0 0 1 | 10 |
| 5 | 0 0 0 1 0 | 11 |
Your mission is to determine the correct combination from the above clues.
Compare the first two clues.
01011
Explanation: Here's one way to solve it: Clues 1 and 2 differ by only the fourth bit, and 1 has one correct bit whereas 2 has two. Thus, the fourth bit of 2, which is a one, must be correct. So, the combination is: _ _ _ 1 _. Similarly, clues 1 and 4 differ only by the last bit, while 1 has only one correct bit and 4 has two. Hence, the last bit of 4, which is a one, must be correct. Therefore, the correct combination is: _ _ _ 1 1. This leaves only the first three digits to be determined, and there are only eight possible permutations of three bits. However, clue 5 tells us that there's only one 1 in the first three digits, since two of its first three bits must be correct for it to have three correct bits. Thus, there are only three possibilities for the first three digits of the combination: 1 0 0, 0 1 0, and 0 0 1. The first trio is ruled out by clues 2 and 4, both of which would have four digits correct instead of only two. The last trio is eliminated by clue 3, which would be entirely correct. This leaves the middle trio. Therefore, the full combination is: 0 1 0 1 1.
* ↑ I, II, III, IV, V, VI, VII, VIII, IX, X, XI.