Boarding an Airplane
By John P. Robertson

One hundred people are boarding an airplane that has exactly one hundred seats. 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, or otherwise take seats at random. What is the probability that the very last person sits in his assigned seat?

Tricky Track Solution
At a track meet, Washington High won with 22 points, and Lincoln High and Roosevelt High tied with 9 points. Lincoln won the shot put. The question was which school won the high jump.

Micah Woolstenhulme's and many other solvers' solutions were as follows. There were at least two events. The total number of points awarded is 40, and the sum of the number of points per event must divide 40. This sum must be at least 6, because 3 + 2 + 1 = 6. So possible point totals for each event are 8, 10, and 20. If the total is 20, first place must be at least 8 points. If first place were 8 points, last place would be 5 points, and Lincoln would score more than 9. If first place were more than 8 points, again Lincoln would score more than 9. If the points per event is 10, there are four events, and possible first, second, and third place scores are (7, 2, 1), (6, 3, 1), (5, 4, 1), and (5, 3, 2). A little trial and error shows that the final total scores above cannot be achieved from any of these. If the points per event is 8, then possible first, second, and third place scores are (5, 2, 1) and (4, 3, 1). If (4, 3, 1), then four firsts and a second would be only 19 points, so Washington could not have scored 22. That leaves (5, 2, 1), and the only way to get the final scores is if Washington wins four events and is second in one event, Lincoln wins one event and is third in the other four, and Roosevelt is second in four events and third in one. So Washington won the high jump.

Michael Belfatti noted that if you assume there is a unique answer, then it must be Washington because the problem is only solvable if one school won all the other events, and that school had to win the meet.

Other solvers include Marty Adler, David Biewer, Jon Evans, George De Graaf, Greg Hansen, Thomas Hess, Todd Hess, Ruth Howald, Steve Kantor, Frank Karlinski, Lawrence Katz, Richard Kollmar, Hoi Leung, Laura Masi, Chrsitopher Mosbo, Timothy Pollis, James Reinbolt, Peter Royek, Dave Schofield, Gregory Scruton, Yipei Shen, David Spiegler, Kathleen Terrill, John Varca, Dave Westererg, Tim Wisecarver, Arlene Woodruff, and Joshua Youdovin.