Ms. Zany's LecturesMath brain teasers require computations to solve.
I am pleased that my little study hall presentations have been so well received. Though few of you have actually seen one, the number of requests for future participation has been overwhelming. (My pleasure would have been greater if my teaching name, Miss Hoehne, had not morphed into "Ms. Zany.")
I want to thank the interns who made it possible for me to escape some of my administrative duties. Without their help, I would not have had the time to proctor an occasional study hall. With their approaching departure, the presentations will come to an end or, at least, an indefinite hiatus.
Next week will be your last chance for a while to witness any of these presentations. All five of the current crop will be recreated and consolidated for recording in Fink College's video studio. (One fifth of a mile from our campus.) It is already clear the studio's size will not accommodate everyone interested and eligible, so the following questions have been made part of the elimination process. (The formal application provides spaces for the answers.)
Twelve items are to be divided into groups of three or four items. The items in the group will be listed, e.g. 1, blue, bench, A. You will guess a target arrangement, e.g. bench, A, 1, blue. You will score one point for each match, e.g. If the target established beforehand is "bench, A, blue, 1," your guess would score two points. To maximize your score, do you choose to guess about three groups of four items or about four groups of three items?
The described test would use our school's computer network, so it will be possible to tell when exactly 250 tests have been completed. Of the first 250 pupils completing these tests, how many will have scored exactly eleven points?
HintDeclaring what has been implied: blanks and repetition are prevented by use of the school's computer system.
AnswerProbability makes four groups of three the higher scorer.
There are 6 ways to arrange 3 items.
Only 1 guess in 6 scores all 3 points contributing 1/2 point to the average.
3 guesses in 6 score 1 point contributing 1/2 point to the average.
The remaining 2 of the 6 guesses contribute nothing to the average.
Each grouping averages 1 point, so 4 groupings will net on average 4 points.
There are 24 ways to arrange 4 items.
Only 1 guess in 24 scores all 4 points contributing 1/6 point to the average.
6 guesses in 24 score 2 points contributing 1/2 point to the average.
8 guesses in 24 score 1 point contributing 1/3 point to the average.
The remaining 9 of the 24 guesses contribute nothing to the average.
Again each grouping averages 1 point, so 3 groupings will on average net only 3 points.
Regardless of which test or mix of tests is taken, the answer is zero. There is no way to get exactly 11 correct, because there is no way to get just 1 wrong.
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