FYS: Puzzles and Paradoxes--Problem Set 4
Complete exactly three of the following five problems.
1. George and Sally both do not have classes on Tuesday and Thursday, so if they leave for Key West on Wednesday after class, skip Friday's and Monday's classes, and come back late Tuesday night, they can have almost thirteen days of spring break rather than nine. Any day they are together in Key West is worth 11 value points to each of them. Any day they are together, but not in Key West is worth 3 value points to each of them. Unfortunately, for one particular course they have, skipping each class comes with a 1% subtraction from the percentage value of their final grades. Suppose for George there is an equal probability of getting any whole number grade from 69 through 84 and for Sally there is an equal probability of getting any whole number grade from 76 through 91. George assigns a -75 point value and Sally a -49 point value to dropping a grade as a result of the late penalty where the relevant grading scale is as follows: D+ = 67.5 - 69.49, C- = 69.5-72.49, C = 72.5 - 77.49, C+ = 77.5-79.49, B- = 79.5- 82.49, B = 82.5 - 87.49, B+ = 87.5 - 89.49, A- = 89.5-92.49.
Assume (a) both George and Sally want to maximize their own individual expected value, (b) that no other values other than the values mentioned are relevant (c) that the only options are 9, 11, or 13 days in Key West for Spring Break, and (d) each is unwilling to go to (and return from) Key West without the other. Is it possible for all four assumptions to be satisfied given the parameters of the problem? Justify your answer.
2. You have been shipped seven tubes of ball bearings. Each tube contains exactly 1500 ball bearings. Each ball bearing is supposed to weigh exactly one gram. Unfortunately, you have learned that in exactly one tube all the ball bearings are defective, i.e. either all weigh four milligrams too much or all weigh four milligrams too little. Your only scale, a digital scale, can weigh no more than 30 ball bearings at a time.
What is the minimum number of weighings that will guarantee you know which tube contains the defective ball bearings? Justify your answer.
3. There are 109 prisoners scheduled to be executed. The prisoners will be lined up, facing the same direction along the line, so that each prisoner can see all the prisoners in front of him or her but none of those behind him or her. A hat will be placed on each prisoner’s head, and each hat has a 50% chance of being red, and a 50% chance of being blue. (The hats are all independent – there aren’t a fixed number of red or blue hats). Starting at the back of the line, each prisoner will be asked by the warden what colour his or her hat is. The prisoner will answer by pushing either a red button or blue button on the remote control the warden carries. If the prisoner gets it right, he or she goes free; otherwise, the prisoner is executed. The prisoners can learn the results of what happens behind them by looking at a large monitor in front of them. The monitor lists the prisoners by number and as the button is pushed either the word 'red' or 'blue' appears next to the prisoner's number in line. The prisoners can see all the hats in front of them. However, they can’t make any noise or any other sort of communication, other than pushing the warden's button, once the hats are placed (or else they will all be put to death). Before entering the line-up they can discuss and agree on a strategy.
What is the maximum number of prisoners who can be guaranteed to survive? Justify your answer.
4. In 1973 women applicants charged the University of California at Berkeley graduate school admissions with sexual discrimination. A study revealed that, despite adequately similar qualifications, 44% of men applicants were accepted, but only 35% of women applicants were accepted. The graduate school, in an effort to determine in which departments the alleged discrimination might be occurring, requested that each department examine its own applicant-to-acceptance percentages. Upon doing so, every single graduate department insisted that, in their department, the probability of gaining admission was better for women than for men.
Can the study and all the departments be telling the truth simultaneously in this situation? Would the added assumption that the number of male and female graduate applications was the same alter the answer to the first question? Justify your answers.
5.[Bonus Point Problem] Dividing Cake: A cake is divided fairly if no one can legitimately complain about the portion they get. A complaint is legitimate if and only if the portion one gets is both (a) not acceptable in one's own eyes and (b) one gets a portion that they were not responsible for creating. Suppose two people want to divide up a cake fairly using just a knife. An obvious solution is to have one person cut and the other person choose which piece to take. The chooser can take whichever piece she thinks is bigger (or smaller if preferred) and certainly cannot complain that she got a portion that is not a acceptable in her own eyes--after all she got to choose from all the possible pieces. If the cutter cuts the cake into obviously different size pieces and the chooser picks the larger piece, the cutter has no one to blame but himself for how he cut the cake and cannot complain that he got a portion that he was not responsible for creating. In the case of three people, however, the same strategy will not work. If the first person cuts the cake into three pieces, and then the second person chooses, the third person may well complain that none of the remaining two peices are acceptable in her eyes. Now suppose there are eleven people who want to divide up a cake fairly using just a knife.
Is there any way to arrange the cutting and choosing amongst the eleven people such that no one can legitimately complain about the portion they get? Justify your answer.