Problem Solving

In a certain game there are 8 steps, referred to as step 1, step 2, and so on with the final step being step 8. The steps are played one after the other. In each step a score of 1, 2, 3, 4, or 5 is obtained. Andrea played the game, getting at least one score of each of 1, 2, 3, 4, and 5, and never getting the same score in consecutive steps. What is the greatest possible score that Andrea could have gotten?

A. 28
B. 29
C. 30
D. 36
E. 40

40 is not possible since each score has to be included once

The next maximizing alternative

5 4 5 4 5 4 5 4 = 36

Not possible since we have to fit in a 1,2 and 3.

Replace three 4's by 1,2 and 3.


I would go with C i.e 30

Since at least one appearance of each from 1, 2, 3, 4, and 5 is the condition, so to maximise the sum of 8 takings, let's take as many 5's as we can, keeping the "no consecutive repeat" in mind, then include the maximum possible appearance of 4's under all conditions. In the end, to ensure that 1, 2, 3 have also been taken once, we get the following possibility that maximises the sum

5 + 4 + 5 + 3 + 5 + 2 + 5 + 1 = 30.

I'll go with C