Charlotte writes a test consisting of 100 questions, where the answer to each question is either TRUE or FALSE. Charlotte’s teacher announces that for every five consecutive questions on the test, the answers to exactly three of them are TRUE. Just before the test starts, the teacher whispers to Charlotte that the answers to the first and last questions are both FALSE. (a) Determine the number of questions for which the correct answer is TRUE. (b) What is the correct answer to the sixth question on the test? (c) Explain how Charlotte can correctly answer all 100 questions on the test.
Problem
Source:
Tags:
jayden94941
18.04.2015 07:23
As 10000th User said, there is a pattern: FALSE, TRUE, TRUE, TRUE, FALSE. It can be easily checked that this satisfies the condition. To prove this is the only solution, we consider the 6th Question. Since from pro.2-5 there are 1 F and 3 T, pro 6 must be a F. This process can be continued to derive that the 11th, the 16th, ..., the 96th questions are all False. Since there must only be 2 false and 3 trues per 5 consecutive questions, the 97th, 98th, and 99th are all True. From this, it is possible to find the 89th, then from that the 88th, ..., which would give us FALSE, TRUE, TRUE, TRUE, FALSE. QED So the number of questions with correct answer true is 60.
geniusofart
18.04.2015 07:42
If you try the $2$nd question as being false, you necessarily find the pattern FFTTT FFTTT FFTTT.... However, this is not correct because the $100$th question becomes true. So, we try the second question being true, and we find that there are many ways. They can be FTTTF FTTTF FTTTF... or FTFTT FTFTT FTFTT... FTTFT FTTFT... but since $100/2$ has remainder $0$, this means that the $100$th question will be one before the first letter. That restriction only works in FTTTF FTTTF FTTTF. So, to answer the questions, (a) $3\times20=\boxed{60}$ (b) True (c) See above.