FORTKNIGHT OF THE YEAR 2012-13: Xidian Sun
THE PROBLEM OF THE FORTNIGHT 2013-2014
· Solutions are due by 4:30 PM on Friday, December 13.
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 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.
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.
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.
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
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