A Drizzle of Puzzles

Today's forecast will include a light rain of puzzles. We are expecting at least 1", but it's harder to determine the upper bound.



Purplified Chessboard (source, with solution)

At time t = 0, some of the squares on my 8-by-8 chessboard inexplicably caught a disease which turned them purple. This disease is contagious: if at time t, a square is adjacent (either vertically or horizontally) to at least two purple squares, then at time t+1, the square becomes overwhelmed by purple-germs and turns purple itself. Once a square has been purplified, it never recovers.

I can tell that my chessboard will eventually become entirely purple. What's the largest possible number of non-purple squares my chessboard can have at time t = 0?




Guessing Game (source)

Aliens have invaded the planet again and chosen you and two other people to determine its fate. Your task is as follows:

• Each of you is given a hat, red or blue, with a 1/2 chance of each color. You can see the colors of the other two people's hats, but not your own.

• Without communicating anything to each other, each of you privately writes down either "my hat is red," "my hat is blue," or "I will not guess."

• If at least one person guesses, and everyone who guesses is correct, the aliens will leave you in peace. Otherwise, they will annihilate humanity.

You can discuss your strategy with the other two people beforehand. If you choose an optimal strategy, what's the probability that the world is saved?

Cutting a Cross (source, with lower bound solution)

You may make two straight-line cuts into the following cross. What is the largest number of pieces you can get?

A Guide to College Math Classes

Hey there! Are you a freshman, just starting out, heart set on majoring in math? A junior who thought she wanted to do computer science but had an unexpected change of heart? A geosciences major who wants to try a math class or two? All are welcome. Sit down, have a cup of tea with me, and I shall bestow my wisdom upon you.

• Find the best professors. Ask older students which ones are good teachers and which ones aren't. Having a good professor can make a huge difference in whether a class is enjoyable or painful.

• If you're allowed to collaborate on p-sets, do that. Having p-set buddies is incredibly helpful, especially in challenging classes. When you get stuck on a problem, they'll help un-stick you, and vice versa. This is a perfectly normal thing to do: mathematicians have an extremely collaborative culture, despite the stereotype of the recluse working alone in an ivory tower. Indeed, in proof-based math classes, working together with your classmates is not only normal, but often essential to doing well in the class.

• Go to office hours if you need help. You've probably heard this before, but that's for a reason. Any decent professors or TAs will be happy to re-explain stuff that you didn't get the first time around. They will not think you are dumb for asking; on the contrary, they'll respect you for putting in the effort to learn. It's much better to be the kid who goes to office hours and asks for help than the kid who doesn't and is eternally confused. If you have a schedule conflict that prevents you from going to office hours, it's totally normal to email the professor or TA to ask to meet at an alternate time.

• Ask questions in class. Admittedly, this depends on your class size; if you're in a 100-person calculus lecture where asking questions feels super awkward, then, well, no pressure. But in smaller classes, definitely go for it. In my experience, professors appreciate being asked questions, and see above re: "what if they think I'm dumb." Plus, if you're confused by something, other students probably are too.

• Sometimes, even if your professor is nice and means well, their style of explanation might not mesh with your style of learning. Or sometimes, it's Sunday and they're not available. Or sometimes, it's Tuesday and you could totally go to office hours, but it's raining and you'd rather stay in your nice cozy dorm. The point is, if you want, you can just ask your classmates for help instead of asking the professor. In my own experience, every time I've asked a classmate to explain something I didn't get, they've been happy to.

• If you've exhausted all other options (or even if you haven't), and there's still a topic you don't really understand, the internet is your friend (as long as you're not using it to look up answers to homework problems). For example, check out my page of math resources. A caveat: the internet has a lot more instructional material for lower-level classes like calculus and linear algebra than for advanced proof-based classes.

• Finally, don't blame yourself if you get lost. It can be really disheartening when you find that your first college-level math class, or your first proof-based math class, is a lot harder than you expected -- but don't beat yourself up. Most college-level math classes are designed to be hard! Be proud: You have taken on this challenge.

And this is the part where I stop, because I have completely forgotten to drink my hypothetical tea.

A Gaggle of Puzzles

What is the collective noun for puzzles, anyway? If there isn’t one, there certainly ought to be.

Here are a few puzzles with simple and thematically similar solutions. They require no math and are accessible to layfolk. I included sources where I could.



Save the World

Aliens have invaded and chosen you and 9 other people to determine the fate of the world. You can strategize beforehand.

• Each person is given a hat, red or blue.

• Person n can see the colors of hats 1, 2, 3, … n - 1 (but not their own). So, person 1 sees nothing!

• In reverse order, beginning with person 10, each person guesses the color of their hat. Everyone can hear what each person guesses.

• People can only say “red” or “blue.” They cannot communicate to each other using pronunciation, speed, etc.

• At the end, for each wrong guess, the aliens kill 10% of Earth’s population.

What is the maximum percentage of humanity that can be guaranteed to be saved?




Mutilated Chessboard (source, with solution)

A standard 8 by 8 chessboard has the lower left corner and the upper right removed, with the remaining 62 squares left intact. You are given 31 dominoes, each of which can cover 2 squares.

• The dominoes may be oriented vertically or horizontally.

• The dominoes may not overlap or go off the edge of the board.

Can you cover the chessboard with the dominoes?



Purple and Orange Polyhedra (source: 2013 HCSSiM Interesting Test)

I have a bin of containing purple and orange plastic polyhedra, 2013 of each. I have plenty of additional polyhedra of both colors, so I won’t run out. Every 17 seconds, I remove two polyhedra at random from the bin.

• If they are the same color, I put a purple polyhedron into the bin.

• If they are different colors, I put an orange polyhedron into the bin.

When there is only one polyhedron left in the bin, what is the probability that it is orange?



Chocolate Bar

You have an n by m rectangular Hershey’s bar, and you want to break it up according to the grid (i.e., in the usual way) to get n*m pieces.

• You can only break one piece at a time; you may not do things like stacking multiple pieces on top of each other and then breaking them.

What are the minimum and maximum numbers of breaks you can use to break up the chocolate bar?