4. In a quiz, you answer 10 questions. Each question is either difficult or easy. For a difficult question, you get 5 points if you answer it correctly and -1 point if you answer it incorrectly; for an easy question, you get 3 points if you answer it correctly and -1 point if you answer it incorrectly. Moreover, if you answer a question correctly, the next one is difficult; if you answer a question incorrectly, the next one is easy. You start with a difficult question.

How many different final scores are possible after 10 questions?
B4. 27 To simplify the problem, we look at a variant of the quiz. We give one extra point for each question: 6 points for a correct answer to a difficult question, 4 points for a correct answer to an easy question, and 0 points for an incorrect answer. Everyone thus gets exactly 10 extra points, so the number of possible final scores does not change. We now see that all scores are divisible by 2, and we divide by 2 to make the problem even simpler. A participant now gets 3, 2, or 0 points for each question. This also does not change the number of possible final scores.

The final score is a maximum of 30 points (if all questions are answered correctly) and a minimum of 0 points (if all questions are answered incorrectly). We will now show that the final scores 1, 25, 28, and 29 are not possible, and that all other scores from 0 to 30 are possible.

The final scores $0,3,6, \ldots, 30$ can be achieved by first answering $0,1,2,3, \ldots, 10$ questions correctly and then answering all questions incorrectly.

The final scores $2,5,8, \ldots, 26$ can be achieved by first answering $9,8,7,6, \ldots, 1$ questions incorrectly, then answering an easy question correctly, and subsequently answering all difficult questions correctly.

The final scores $4,7,10, \ldots, 22$ can be achieved by first answering $7,6,5, \ldots, 1$ questions incorrectly, then answering an easy question correctly, a difficult question incorrectly, another easy question correctly, and all difficult questions correctly thereafter.

If at least one question is answered correctly, you already have at least 2 points. For this reason, the final score of 1 is not possible. If at least one question is answered incorrectly, you have at most $9 \cdot 3=27$ points. For this reason, the final scores of 28 and 29 are not possible. If exactly one question is answered incorrectly, two things can happen: the incorrectly answered question is the last question and you have $9 \cdot 3=27$ points, or the incorrectly answered question is not the last question and you have $8 \cdot 3+2=26$ points. If at least two questions are answered incorrectly, you have at most $8 \cdot 3=24$ points. For this reason, the final score of 25 is also not possible.

We see that the possible final scores (in the new quiz format) are the numbers from 0 to 30, with the exception of 1, 25, 28, and 29. (In the original quiz format, the possible final scores are the even numbers from -10 to 50, with the exception of $-8,40,46$, and 48.) There are thus 27 possible final scores.