The Problem of the Fortnight

2015 -- 2016

The Problem of the Fortnight is a bi-weekly contest problem offered by the Department of Mathematics and Computer Science. Anyone in the wider Wabash community may submit solutions: students, faculty, staff, and alumni. Submissions will be judged on their correctness and elegance.  Authors of correct solutions will be acknowledged, and the best solutions will be posted. Repeat solvers win prizes! The student who solves the most problems in an academic year is declared FortKnight of the Year.


FortKnight of the Year for 2016—2017 is Yang Yang

  Four years in a row!






Answers to Previous Problems                                                                 Problems from 2015-2016


Fortnight Problem #12.  Suppose you have an incorrect two-pan balance.  You can put weights on the pans and get them to balance, but the weights aren't equal.  Some particular ratio of the weights will make it balance, but you don't know what that ratio is.  Suppose you have a 5-pound weight, a 6-pound weight, a pile of sand, and a collection of weightless containers you can put sand into.  Explain how to measure 1 pound of sand.  If you have a 7-pound weight instead of the 6-pound weight, can you still measure 1 pound of sand? (Thanks to Professor Gregory Galperin of Eastern Illinois University for suggesting this problem.)

I received correct solutions from Joseph Bertaux, Benny Liang, Kevin Sheridan, and Yang Yang. Kevin's solution is the easiest to understand, but none of the solutions were as simple as the one I had in mind. A correspondence with Professor Morillo made me realize that there are two ways that the balance might be incorrect. One is if the fulcrum (balance point) is not halfway between the pans.  In this case weights that balance each other are in some constant ratio relative to each other. The second way is if one pan is heavier than the other. In this case weights that balance each other have a constant difference. 

Here is my solution, which works for both types of incorrect balance.   Put the 6-pound weight on one pan and put an amount of sand on the other pan to make it balance.  Replace the 6-pound weight with the 5-pound weight plus enough sand to make it balance again. The additional sand with the 5-pound weight is one pound.

Given two known weights, this process will result in an amount of sand the difference of the two weights.  Thus, starting with 5-pound and 7-pound weights, we can measure 2 pounds of sand.  Do this twice and combine them and we have 4 pounds of sand.  Treating this as a 4-pound weight, we can then use it and the 5-pound weight to measure one more pound of sand.

Fortnight Problem #11. Consider two non-intersecting circles in the plane. From the center of each circle draw the tangents to the other circle as shown. These tangents determine chords AB and CD, shown in red. Prove that these two chords are congruent.  For extra kudos, prove this is true in spherical and hyperbolic geometries.   I received correct solutions from Joseph Bertaux, Ngoc Tran, and Yang Yang.  Yang's solution includes spherical and hyperbolic geometries.


Fortnight Problem #10.   I hope you are enjoying spring break.  Some of you may be sitting by the pool or playing pool. If the latter, please consider the following billiards problem. Billiards, similar to pool, but without pockets, is played on a table that is 10 feet by 5 feet. In a simplified version there are two balls, the cue ball and the target ball.  The player attempts to have the cue ball hit specified cushions before hitting the target ball.  Imagine that the table is oriented in the east-west direction. The cue ball is halfway between the north and south cushions, x feet from the east cushion. The goal is to hit the target ball, wherever it might be, after first hitting the south and west cushions, in that order. Evidently this is possible for some target positions and not for others--there will be some region of the table where the target ball might be for which such a shot is possible. Your task is to find the area of this region as a function of x.  If it is difficult to find this as a function of x, find it for a few values of x, say, when the cue ball starts 2 feet, 4 feet, or 6 feet from the east cushion. Note: If the cue ball is shot directly into a corner, it rebounds in the same direction it came from.

               
Possible                                                                            Impossible

I received correct solutions from Joseph Bertaux and Yang Yang.  Joseph included a neat Mathematica animation showing the region where the target can be.  Here is Yang's solution.  Here are two images from Joseph's animation.  If the cue ball starts on the right half of the table, the region where the target ball can be is a trapezoid (shaded blue).  If the cue ball starts on the left half of the table, the region is a triangle.




Fortnight Problem #9


I received correct solutions from Joseph Bertaux, Meng Fan, and Yang Yang.  Here is Yang's solution.



Fortnight Problem #8.  Suppose that a chicken and a half can lay an egg and a half in a day and a half. What combination of chickens and days will give you 14 eggs?  No fractional chickens or days, please!

Seven chickens in three days is one of the combinations, and there are at least two more.  I received correct solutions from Joseph Bertaux, Craig Knoche, and Yang Yang.   Coach Knoche's solution is the most straight forward.   Joseph gets the prize for the most exhaustive solution (or perhaps the most exhausting)! He managed this by carefully examining the exact meaning of the hypothesis of the problem. (One sentence was highlighted by the editor.)


Fortnight Problem #7.  A fast mouse runs along the x-axis in the positive direction with speed m. A fat cat is sitting at the point (0,C).  When the mouse is at the origin the cat sees him and decides to chase. The cat runs with speed c, which is slower than the mouse.  What path should the cat take so that he comes as close as possible to the mouse?  

https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcS6kijr8gCfJCF0HwoS2K8UwFESMqdFPc_tybxRCxK_9Ruzz7jv
No correct solutions were received by the deadline.  After giving some hints, I received a correct solution from Ngoc Tran. The cat should run towards the point on the x-axis with coordinate x = Cc/(m^2 - c^2) .  See his solution here.


Fortnight Problem #6.  Crawfordsville lies at a latitude of 40 degrees north.  Consider the circle of all locations on earth with this latitude. Explain why, at any given moment, there must be two locations on the circle that are opposite each other and that have the same temperature.


This is the Intermediate Value Theorem in disguise!  I received correct solutions from Kevin Sheridan and Yang Yang.  Kevin's solution was particularly well-presented.


Fortnight Problem #5.  Suppose P1 and P2 are distinct points in the plane and that L1 and L2 are distinct, non-parallel lines through P1 and P2. Describe how to locate a point C that will be the center of a rotation that takes P1 to P2 and L1 to L2. Your description need not be a compass and straightedge construction, but you should, of course, include an explanation of why it works.

I received very different correct solutions from Joseph Bertaux, Kevin Sheridan, and Yang Yang.  Joseph and Yang pointed out that there are two points you can rotate about.  Joseph also noted that the lines can be parallel. In that case you can rotate 180 degrees around the midpoint of P1 and P2.  Here is Kevin's solution.


Fortnight Problem #4. An ant is at one corner of a 4x4x4 Rubik's cube. It wants to crawl to the opposite corner along the edges between the colored squares. How many paths of minimal length can it choose?


Believe it or not, there are 2550 paths the ant can take to cross the Rubik's cube.  I received correct solutions from Joseph Bertaux, The Anh Pham, Kevin Sheridan, and Yang Yang.  Here is The Anh Pham's solution.


Fortnight Problem #3.  In honor of the 50th anniversary of Star Trek, we go where no fortnight problem has gone before!  Kirk and Spock beam down to an alien planet, and they materialize in what appears to be a school room.  They see the following arithmetic facts on the blackboard:

13 + 15 = 31             10 x 10 = 100                       6 x 3 = 24

The alien children are returning from recess. How many fingers do they have?

The aliens are doing arithmetic in base seven.  Thus they likely have either seven or fourteen fingers. Most of the solvers said seven fingers; a few thought it would be strange to have different numbers of fingers on each hand; only two considered fourteen fingers. (There are a few human cultures that count in base five.)  We had lots of solvers this time: Joseph Bertaux, Jim Brown, Matt Hodges, Craig Knoche, Kevin Sheridan, Ngoc Tran, Brad Weaver, Yang Yang, and Jonah Woods.   Most gave proofs that the base has to be seven. A few showed that base seven works, but didn't argue that it is the only base that works.  Here is Matt Hodges's proof.


Fortnight Problem #2.  An Astonishing Annuity. Given interest rate 5%, what is the minimum amount you need to deposit in an account so that you can withdraw $1 at the end of the first year, $4 at the end of the second year, $9 at the end of the third year, and in general, $n2 at the end of the nth year?  You may assume that the interest is compounded annually, quarterly, monthly, daily, or even continuously (but be sure to make it clear which).

If you initially deposit $17,220.00, you can withdraw this increasing amount forever!  I received correct solutions from Joseph Bertaux, Craig Knoche, and Yang Yang. Joseph and Yang submitted proofs. Coach Knoche submitted a very cleverly programmed spreadsheet. It illustrates that if the initial deposit is $17,220.00, the principle continues to grow enough to make this increasing annual withdrawal. However, if the initial deposit is $17,219.99, the fund runs out of money after a few hundred years!  Here is Joseph's proof.


Fortnight Problem #1. Suppose that a circular road could be built centered at Indianapolis with a radius of 500 miles.  What would the length of the road be? (If you think this is trivial, then you haven't thought about it carefully!) Can you generalize this?

The answer is approximately 3133 miles, which is slightly less than what you get with 2πr,  due to the curvature of the earth.  \piI received correct solutions from Joseph Bertaux, Ethan Hollander, Manzil Mudbari, Kevin Sheridan, and Yang Yang.  Three of the solvers generalized their answers to circular roads on planets of arbitrary size.  Yang Yang's solution was very nicely written.