**FORTKNIGHT
OF THE YEAR 2012-13: Xidian Sun**

**THE
PROBLEM OF THE FORTNIGHT 2013-2014 **

**Problem 7**

- 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, December 13.

**Problem 6**

Given
a circle, six distinct points are chosen at random on the circumference and all
six points are connected to each other by line segments which are randomly colored
either blue or red. What is the probability that three of the points form a
triangle with all its sides being of the same color? (That is, what is the
probability that either a red or a blue triangle is created?)

The probability is 100%. This result is a classic one from Ramsey
Theory. Many interpretations of the result can be made. For example, at any party with at least six
people, there are three people who are all either mutual acquaintances (each
one knows the other two) or mutual strangers (each one does not know either of
the other two). I received complete solutions from Davis Stone, Yang Yang and
Zhipu Ye.

The neatest solution was David Stone’s, which is here.

**Problem 5**

Problem 5 was solved by Zhipu Ye, Cameron Dennis,
David Stone, Yang Yang and Ngoc Tran. Two partial
solutions were received.

In this judge’s opinion, the most elegant solution
is Ngoc Tran’s, which you will find by clicking here.

**Problem 4**

Solution:
The function is not periodic. The following submitted correct solutions: Ngoc
Ngo Quang Tran, Zhipu Ye, Yang Yang, Cameron Dennis, Korbin West and Xidian
Sun.

Click
here for
Xidian’s solution.

**Problem
3**

Wally and his spouse invited four other couples to a
cocktail party. As the guests arrived and greeted each other, a few handshakes
took place. Of course, no one shook hands with themselves and no one shook
hands with his or her spouse. Being a numbers person, Wally asked everyone to
tell him how many hands they had shaken. To his surprise, he got nine different
answers! How many hands did Wally’s spouse shake?

Solution: Wally’s spouse shook 4 hands. Successful
solvers of problem 3 were David Stone (’91), Ngoc Ngo Quang Tran, Aaron
Wirthein, Cameron Dennis, Zhipu Ye, Korbin West, and Yang Yang.

Click here for
Cameron Dennis’ solution and here for Zhipu
Ye’s solution

**Problem
2**

What is the sum of all the digits in the integers
from 1 to 1,000,000?

What is the sum of all the digits in the integers
from 1 to 10,000,000?

What is the sum of all the digits in the integers
from 1 to 100,000,000?

(To clarify: the sum of all the digits of the
numbers 23 and 436 is 2+3+4+3+6=18.)

Problem 2 was solved by Ngoc Tran, Rob Barber,
Albert Li, Dave Stone, Jacob Eliot Nettnay, Yang Yang, Zhipu Ye, Cameron
Dennis, Brad Weaver and Korbin West.

Click here for Rob’s
solution

**Problem
1**

Going at top speed, Indy 500 driver X leads his
rival by a steady three miles. Only two miles from the finish line, X runs out
of fuel. Thereafter, X deceleration is proportional to the square of his
remaining velocity and, in the next mile, his speed exactly halves. Who wins
and why?

Problem 1 was solved by Cameron Dennis, Korbin West,
Yang Yang and Zhipu Ye.

Click here for
Korbin’s solution