A Mathematical Zoo: A Wild Puzzle Compilation

Updated: Sep 12

Last month, we looked at one of my favorite investigative problems—involving coconuts, pirates, and of course, thieving monkeys—diving into the jungle to uncover fascinating arithmetic that can be traced back to the ancient Greeks. For July’s in-depth monthly edition, I bring you a companion article of sorts: I'll pose several more animal-themed puzzles to you to try. You won't want to miss the thrill of the chase on these wild puzzles, so go get some scrap paper and a pencil, and enter the zoo once again with me!

To add a little bit about my inspiration for this issue, I've noticed how much you all loved my previous puzzle compilation articles, so it seemed like the perfect time to bring that back. Here are a few previous compilations if you’d like to check them out:

Just like my previous puzzle compilations, I’ll place the puzzles in the main section of the article below and you can find solutions at the end. Note that some of the solutions are entirely complete, but others may guide you to find the answer yourself; in each case, I chose the approach that prioritizes the most enlightening process of discovery so you can get the most out of this article. Some of these puzzles are fairly easy on the surface but are actually much more complex, so make sure to read the in-depth solutions to get the most out of the process!

As you might imagine, animals are rather a broad and popular topic across the board in the math world, so it was quite difficult to narrow this article down to just these puzzles. I worked hard to make sure the puzzles I decided to include are accessible to practically everyone. There are so many amazing questions out there, so if you find any others that are particularly exciting, you are always welcome to share them with me and GLeaM in the forum.

1) Cats vs Rats

Source = Entertaining Mathematical Puzzles by Martin Gardner

Our first puzzle will serve as a warm-up. It’s unclear exactly where this puzzle originated because it’s been quoted so many times across the internet; it turns out it is even used frequently as a job interview question. It is, however, featured in one of Martin Gardner’s books, so that is the most-agreed upon source. Regardless, it’s a great example of when jumping to conclusions can be misleading.

If three cats can catch three rats given three minutes, how many cats would catch a hundred rats given a hundred minutes?

Here’s a bit more advice: The answer might not be the first one that comes to mind, but it also might not strictly be the second one either… you will want to think about proportions, but you will also want to think about the way cats catch rats in the first place. Is there really a good answer at all?

When you have an answer, go ahead and scroll to the end of the article!

My cats, Jasper and Coco, who would catch exactly zero rats even given a hundred years.

2) Jumping Frogs

Source = So You Think You’ve Got Problems? by Alex Bellos

This is yet another problem that has much more to it than initially meets the eye.

Fred the frog is sitting on the first lily pad in a line of lily pads. He wants to reach the tenth lily pad, and he can either jump to an adjacent lily pad (one down the line) or jump over an adjacent lily pad and land on the one after it (two down the line). If he never moves backwards, in how many ways can Fred reach the tenth lily pad?

If you’ve been in the math world for some time, it’s quite likely that you’ve seen many problems involving frogs jumping on lily pads. It is, after all, one of the easiest ways to set up a sequence or coordinate geometry problem in a more interesting and exciting way.* If so, this one may seem simple in comparison, but we’ll be able to take it much further than you might initially realize.

As you’re solving the problem, ask yourself the following further questions: What answers would we get if Fred wanted to reach the eighth lily pad? The ninth? The eleventh? The twelfth? Is there any pattern you can find? Could you find an expression for the answer if Fred wanted to reach the nth lily pad?

We’ll discuss this more in the solutions, but you can actually find the answer to Fred’s problem in two ways: recursively and explicitly using combinations aka choose functions. By setting these two solutions equal, we are going to prove a rather fun fundamental identity in combinatorics—all from a silly little lily pad problem! (If you’re ambitious, feel free to see if you can figure out what I mean before skipping to the solutions.) This is my favorite solution discussion, so make sure you read it!

Fred resting before his journey. Courtesy of Adobe Stock.

*If you don’t believe me, here are just a couple recent AMC problems involving frogs. If you want to solve any of them, simply click on the title of the problem and it will take you to the solution.

2020 AMC 10A #13/2020 AMC 12A #11

2010 AMC 12B #18

2014 AMC 10B #25/2014 AMC 12B #22

3) Ten Canaries

Source = So You Think You’ve Got Problems? by Alex Bellos

Originally phrased in terms of rats, this is one of my favorite puzzles to grasp the sheer concept of scale—and (hint!) the power of powers. The following is very close to a verbatim restatement of the puzzle, so all credit goes to Alex Bellos.

You have inherited a collection of one thousand bottles. All the bottles contain wine except one, which contains poison. The only way to discover what’s in a bottle is to drink it. If you drink poison, however, you die.

Thankfully, you have 10 canaries. If a canary sips any amount of poison, or poison mixed with wine, it will die after exactly one hour. If a canary drinks just wine, it survives. How do you determine which bottle is poisoned exactly one hour after the first canaries are given their first sips?

You should be able to come up with many schemes given unlimited time, but you only have one hour. What should you do?

Here is a hint: You are allowed to mix small amounts of any number of bottles together and give each canary a mixed concoction to sip from at the same time. That way, you will be able to gain information from the dead canaries at exactly one hour; all the dead canaries will have sipped from a concoction that contains the one poisoned bottle. However, you will only have ten pieces of information. Which bottles should you have each canary sip from to identify the exact poisoned bottle out of a thousand bottles with only ten canaries? What scheme should you follow?

Two canaries. Courtesy of BioMed Central.

4) Fox and Goose