Guess the Rule
By John P. RobertsonThe problem is to guess the rule and fill in the empty circle (see accompanying diagram). Note that the 7 at the bottom is not a misprint. As a hint, I will tell you that the answer is not 15
Solution to "Boarding An Airplane"
The puzzlement was as follows. One hundred people are boarding an airplane that has exactly one hundred seats, and each person has a seat assignment. The first person to board forgets her seat assignment, and just takes a seat at random. Subsequent passengers take their assigned seats, if they are free, and otherwise take seats at random. What is the probability that the very last person sits in his assigned seat?Timothy Pratt's solution is that the probability is 50 percent. He considered the equivalent problem: instead of having passengers whose seats are occupied find new seats, have the person in the wrong seat, who will necessarily be the first person to have boarded, take another seat at random. If the first person eventually sits in her assigned seat, then the last person will get his assigned seat. If the first person eventually sits in the last person's seat, then the last person doesn't get his seat. The first person is equally likely to sit in her own or the last person's seat, so the odds are 50-50.
Many solvers used a recursive formula to deduce the probability when there are 100 seats from the probabilities when there are fewer seats. When there are at least two seats, the probability is always 50 percent.
Other solvers are Avraham Adler, Alan Clark, Jon Evans, George De Graaf, Mark Gillam, Greg Hansen, John Hinton, Shiwen Jiang, Rob Kahn, Alvin Lai, Charles McClenahan, Christopher Mosbo, Brian Mullen, David Oakden, Mitchell S. Pollack, Steven Sousa, Rajesh Thurairatnam, Edward Tyrrell, David Uhland, Humberto Valdes, Dave Westerberg, Tim Wisecarver, Micah Woolstenhulme, Walt Wright, and Bernard Yoo.
Tricky Track
Unfortunately the names of two solvers were omitted in the last issue. Eugene Shevchuk and David Uhland also solved this problem.
