Can you solve this diabolical combination lock problem?
A lock has four buttons and is opened by a sequence of 6 button presses. Pressing the same button a second time in a row has no effect, which is to say that the combinations cannot have two identical numbers in a row. If we are told that the first and last numbers of the combination are both button 1, how many possible combinations are there?
Small numbers were deliberately chosen to increase the frustration level. You might initially think that since, for each of the 4 unknown numbers, there are 3 numbers different from the preceding number, that the answer is 3×3 x 3×3 = 81. However, the last of the 4 numbers cannot be 1, because of the way the lock works, so the answer is some number less than 81. No fair guessing. You must provide an explanation.
Hint: Try solving the problem if the total number of presses was 3 and then try for 4.