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.
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.
Solved
by Xidian Sun. Here is his solution.
Solved
by Xidian Sun and David Stone ’91.
Click here for Xidian Sun’s Solution
