FORTKNIGHT OF THE YEAR 2011-12: Anh Tran
THE PROBLEM OF THE FORTNIGHT 2012-2013
· Solutions are due by 4:30 PM on Friday, April 19.
Problem 13 was solved by Xidian Sun.
Click here for his solution.
Pick any two (they will have more than “1” thing in common):
Recall that a natural number bigger than 1 is prime if its only factors are 1 and itself. Otherwise it is composite. Show that in any collection of 15 composite numbers selected from the first 2013 natural numbers, there have to be two having a common factor bigger than 1.
Solved by Cameron Dennis, David Stone, Steven Kidd, Brad Weaver and Xidian Sun.
Click here for Cameron Dennis’ solution
Two Puzzles About Liars:
In the fictional island of Truth Or Truthiness, there are two types of creatures: Knights, who always speak the truth, and Knaves, who always lie.
Puzzle # 1:
A visitor to their island meets a party of three islanders and asks one of them: "What are you?" The first one mumbles an answer, which the visitor cannot understand. The second says: "He said he is a Knave." The third says to the second: "You're a liar" Is the third in the group a Knight or a Knave?
Puzzle # 2:
In the same situation as above, two of them speak thusly:
A says: “B is Knight”
B says: “A is not a Knight”
Prove that one of A or B is telling the truth, and he is the island's visitor.
Problem 11 was solved by Korbin West, Jacob Scherb, Bryan Tippmann, Xidian Sun, David Stone and Brad Weaver.
Click here for Korbin West’s solution
Stacking the odds:
You have two buckets full of marbles: one bucket contains 100 red marbles, the other has 100 pink marbles. Your friend is going to pick a marble at random from one of the buckets, after the bucket has been well shaken so that the marbles are randomly distributed inside the bucket. If she picks a red marble, she wins $5 from you. If she gets a pink marble, she gives you $3. You have five minutes to set everything up. Is there a way to make this game work in your favor?
Solution: Yes, put one pink marble in one bucket and the 199 remaining marbles in the other bucket.
Correct solutions were submitted by Bryan Hutchens, Zachary Vega, Jacob Scherb, Brad Weaver, Dave Stone, and Xidian Sun.
Click here is Mr. Scherb’s Solution
Solved by Colin McClelland, David Stone and Xidian Sun. Click here for Colin’s solution.
(Note: One can easily show that b = 0. After that, the equation turns out to be equivalent to Cauchy’s functional equation f(x+y) = f(x) +f(y), and “standard” solutions to this problem are easily found around the Web. I have posted Colin’s solution, because I had never seen that novel approach before. Thank you Colin!)
Solution: The function must linear. This problem was solved by Xidian Sun and Dave Stone.
Solved by Xidian Sun, David Stone and Colin McClelland.
Problem 6 was solved by David Stone and Xidian Sun.
Which (if any) of the following sentences are true? Explain your reasoning.
1. Exactly one of these ten sentences is false.
2. Exactly two of these ten sentences are false.
3. Exactly three of these ten sentences are false.
4. Exactly four of these ten sentences are false.
5. Exactly five of these ten sentences are false.
6. Exactly six of these ten sentences are false.
7. Exactly seven of these ten sentences are false.
8. Exactly eight of these ten sentences are false.
9. Exactly nine of these ten sentences are false.
10. Exactly ten of these ten sentences are false.
Solution: Sentence number 9 is the only one that is true.
This problem was correctly solved by the following : Timothy Locksmith, Ben Sia, Matt Michaloski , Corey Hoffman, Robert Christopher Dixon, Charles Wu, Xidian Sun, Rodrigo Alejandre, Korbin West, Nazir Tokhi, David Gunderman, Nash Jones, Patrick Carter, Luke Holm, Mahlon Nevitt, Jon Daron, Jiaxi Lu, Matthew Dickerson, Ryan Horner, Carl Rivera, Jacob Sheridan, Zack Sticher, Tyler Peterson, Adam Togami, Adam Barnes, Dave Stone, Brad Weaver, and Brandon Dothager,
Solved by Xidian Sun. Here is his solution.
Prove that for all positive integers n the equations x2+y2 = 2n and x2+y2 = n have the same number of integer solutions
Solved by Xidian Sun and David Stone ’91.
Solved by Xidian Sun and Colin McClelland ’06.
You are on the roof of an 80 feet high building and you need to get to the ground without going inside the building. You have a rope 60 feet in length which is just strong enough for your weight, and you may not jump or fall. There are no windows or doors, the only way down is along a vertical wall with two solid rings to which you can attach your rope, one at the edge of the roof and the other at the midpoint (40 feet from both roof and ground). You can cut and tie the rope as much as you like, disregard the length of the rope you spend on knots. Can you get to the ground?
The “mathematical” solutions were found by Adam Barnes, Brad Weaver, James Jeffries, Jim Cherry (Sr.), Mark Shaylor, Dave Stone, Jim Brown and David Gunderman. Other solutions involving special knots or maneuvers with the rope were also correct, and were found by A. J. Metz, Will Oprisko and Matthew Dickerson.