**FORTKNIGHT
OF THE YEAR 2011-12: Anh Tran**

**THE
PROBLEM OF THE FORTNIGHT 2012-2013 **

**Problem 14**

- Submit solutions to Esteban Poffald either by email
(poffalde@wabash.edu) or drop them by his office in Goodrich 211.
- Repeat solvers win prizes.
- You do not have to be the first to submit a correct
solution; you just need to submit a correct solution by the deadline.

·
Solutions
are due by 4:30 PM on Friday, April 19.

**Problem 13**

Problem
13 was solved by Xidian Sun.

Click
here for his
solution.

**Problem
12**

**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

**Problem
11**

**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

**Problem 10**

**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

**Problem 9**

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!)

**Problem 8**

Solution: The function must linear. This problem was
solved by Xidian Sun and Dave Stone.

**Problem 7**

Solved by Xidian Sun, David Stone and Colin McClelland.

Problem 6 was solved by David Stone and Xidian Sun.

**Problem 5**

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 **B**randon
Dothager,

**Problem
4**

Solved
by Xidian Sun. __Here is his solution__.

**Problem
3**

Prove
that for all positive integers *n* the
equations *x*^{2}+*y*^{2} = 2*n* and *x*^{2}+*y*^{2} = *n* have the same number of integer solutions

Solved
by Xidian Sun and David Stone ’91.

__Click here for Xidian Sun’s Solution__

**Problem
2**

Solved
by Xidian Sun and Colin McClelland ’06.

**Problem
1**

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.