Math Puzzle No. 4: The Hungry Mouse

By Anonymous

Imagine a block of cheese has been cut into thirds along each face. A friendly neighborhood mouse comes along and starts eating one of the corners. The mouse will eat the entire block of cheese, always progressing to an adjacent cube using only up, down, or cardinal directions (North, South, East, West;  sorry, no diagonals!). 

Challenge: Can you find a path following these rules so that our mouse friend finishes by eating the center cube of cheese last?

Hint: A diagram showing the block cut into cubes -- and another showing the faces made up of cubes --- may help you visualize the problem.

Update: 1/25/21: read below for the solution.

Upon initial inspection, we would likely begin solving a problem like this by taking the grid and trying to find a route for the mouse to eat all of the cheese, somehow ending on that sweet, delicious, final center cube. It is so tempting to just sit and spend a few hours mindlessly drawing grids and different routes through them. However, after a few hours of near successes, (and our friends looking at us like we're going a wee bit mad), we might start to suspect that such a path is impossible! But now we have a new problem! Short of writing down every imaginable route, how would we prove that our poor mouse friend is out of luck and that she'll just need to eat her cheese out of order and maybe she should see a professional to help manage her OCD tendencies.

It turns out, that we can solve this problem for her in just a few sentences by introducing a bit of artificial information! Let's start by labeling the corners where the mouse could start, and then labeling only those squares the mouse could get to after 2 moves with dots, as below:
Now, we can see that there is a total of 27 cubes of cheese for the mouse to eat: 14 with a dot, the other 13 of them dotless. According to our mouse's preference, she will always begin on a dotted cube, and move back and forth between dotless and dotted. After 26 cubes of cheese have been consumed, our dear friend mouse will have eaten 13 with dots, and 13 without. Oh no! We are always then left with a dotted cube, and the center cube was dotless! Sadly, this means we must quietly take our good friend mouse (with whom we have spent many hours trying to solve this problem) aside and let her know she needs a new approach. Has she considered taking up peanut butter?




The previous Math Puzzle and its Solution are available here: https://mathcep.blogspot.com/2020/11/math-puzzle-no-3-ten-divisibilities.html

Comments

Post a Comment

Popular posts from this blog

Mathematically-Themed Icebreakers 2: Hiding a Favorite Theorem in a Classroom Icebreaker

By Anonymous
Image
This week, we discuss an activity that arose by turning a favorite theorem (in this case, a result from Ramsey Theory ) into a first-day activity that got students flexing their mathematical muscles right from the start of class.   This Icebreaker was created by one of our post-docs: Kris Hollingsworth. This activity works well both in the classroom and over Zoom breakout rooms; in fact, it is the first activity we designed from scratch for the remote teaching environment. This activity has been used in an Introduction to Proofs class with 160 students with great success, but would be appropriate in almost any mathematics course. We'll first describe the activity and how we would expect it to work (and maybe you can play along to try and solve the math as well!).  Then, we'll prove the underlying theorem in a way that can be done in class and will finish with some possible suggestions for follow-up discussion. 1) Start by breaking the class into small groups of 4-5 students eac
Read more

MathCEP Director Receives Wolfram Research Innovator Award

By
Image
I was honored earlier this month to be recognized by Stephen Wolfram as one of the recipients of the Wolfram Innovator Award at the annual Wolfram Technology Conference.   (Here is the official announcement of the Award Winners . ) This award is given annually to educational and scientific pioneers that use Wolfram’s Mathematica ™ software to bring the future to today.   The awarded project was the result of a large number of people at the University of Minnesota working to lower the cost of college attendance by replacing expensive textbooks with technology better suited to the needs of 21 st century students. In addition, this technological innovation has improved the way we deliver instruction at MathCEP . The project, now known as the Minnesota Online Learning System (MOLS), delivers assessments to students using the Mathematica platform. It replaced a publisher’s automated homework system initially, and then expanded to house the University of Minnesota’s math placement exam.
Read more

Math Puzzle No. 5: The Damsel in Shining Armor

By
Image
Long ago, damsels in shining armor battled dragons and rescued knights in distress on a daily basis.  This ultimately lead to the complete extinction of dragons, a major catastrophe for the business of thrilling heroics! Although many of the surviving stories don't stress it, the rescues frequently involved logical thinking and problem-solving. Once upon a time, a knight was captured by an evil dragon and imprisoned in a castle that was surrounded by a square moat. The moat was infested with extremely  hungry and alligators, piranhas, and -- just for good measure -- a few sharks.  The moat was exactly 20 feet across and had no drawbridge or any other means of safely crossing. After a long time, a damsel appeared, carrying two wooden beams, each 19 feet 8 inches long and 8 inches wide. The beams were strong enough to walk across, but wouldn't reach across the moat.  The knight was despondent.  He'd already waited for weeks to be rescued.  Now he'd have to wait more, unti
Read more