THE PROBLEM OF THE FORTNIGHT (SEASON 09-10)

Problem 6

 

(From John Maharry, via David Maharry)

 

Spreading Disease on a Chessboard:

 

There are n2 people on an n-by-n chessboard. Some of them are "infected". At each step, anyone who is infected remains infected, and any healthy person with at least two infected neighbors (corners do not count as neighbors) becomes infected in the next step. Notice that if the main diagonal starts out infected, then after n-1 steps everyone is infected.

 

Prove that at least n squares must be infected initially to eventually infect everyone.

 

Solutions are due by 4:30 PM on Friday, November 20.

 

 

(Scroll down for previous problems and solutions)

 

 

 

 

 

 

 

 

Problem 5

Losing your hat, not your head:

You are paddling your canoe upstream at a constant velocity. After paddling for six miles, the wind blows your hat into the stream and the hat begins flowing downstream. You continue to paddle upstream for two more hours before noticing that your hat is missing, at which time you turn around and paddle downstream at the same rate you had paddled upstream, overtaking your hat just as you return to your original starting point. What is the speed of the current?

Answer 1.5 miles per hour

Correctly solved by  Chris Beard, Tianren Wang, Yifei Sun, Kessler Armbruster, Kody Lemond and Robby Dixon.

 

Solution by Robby Dixon

The winner this week is Yifei Sun.

Solution by Yifei Sun

 

 

Problem 3

(From David Maharry)

A hundred people board a fully booked aircraft.  Unfortunately, the first person in line somehow loses his/her boarding pass while entering and takes a random seat.  Each successive passenger then sits in his/her proper seat, if available; otherwise, each one rather wimpily takes a random vacant seat.  What is the probability that the last passenger finds the properly assigned seat unoccupied?

 

Answer: The probability is ½ (regardless of the number of seats, as long as there are at least 2 seats).

 

Correct solutions were given by Ben Burdett, Tianren Wang, Samer Kawak and Yifei Sun.

 

 

Problem 2

Asphalt Savings:        Four towns lie at the corners of a ten-mile square. In order to improve communications between the towns, the towns officials decided to build roads linking all four towns together. Because of a shortage of money for construction projects, the engineers in charge of designing the roads were charged with the task of building a road system that would be as short as possible and still allow access from any one town to any other. What is the design that the engineers should propose?

Answer: The shortest road has length 20√2 ≈27.32 miles.

 

Correct solutions were submitted by Kody LeMond, Bryan Hutchens, Tianren Wang, Ben Burdett, Yifei Sun, Brad Weaver and Tom Runge. Wang, Sun and Weaver also included a proof that that 27.32 miles is the shortest possible total length of the roads:

Solution by Yifei Sun

Solution by Brad Weaver

 

 

 

 

 

 

Problem 1

The Five Fair-Minded Monkeys:       There was a pile of bananas on the beach belonging to five monkeys and their plan was for each to take an equal share. The first monkey came to get his share, but after waiting a long time for his mates, decided to take his share and leave. He found that dividing the number of bananas by five would leave one extra banana, so he threw it into the sea, took one-fifth of the remaining bananas and left. When the second monkey came, he faced the same situation: he wanted to take one-fifth of the bananas, but to do that evenly, he had to throw a banana into the sea. Then, he took what he thought was his share and left. Later, one by one, each monkey came and took what he thought was his share by doing the same as the first two monkeys. What is the least number of bananas in the pile at the beginning? What is the least number of bananas left on the beach after all the monkeys take theirs?

 

Answer: The least possible number of bananas at the beginning is 3121, and at the end 1020 bananas are left.

Correct solutions were provided by Adam Fritsch, Richard Dent,Yifei Sun, Tianren Wang, Bryce Shellman, Kody LeMond,Jacob Eliot Nettnay, Jeromy Troy Sisk, Brad Weaver, Ben Burdett, Conor Frame, Jordan Jeffrey Hoerr and Ryan Cronin.

Solution by Adam Fritch

Solution by Tianren Wang

 

 

Problems from previous years