2024 in math puzzles. – Math with Bad Drawings

2024 was a good year to do math puzzles. Can you guess why?

Easy: because the last 100,000 years have all been good years to do math puzzles.

But this year was especially good, because it saw the publication of (at least) three fabulous new books of mathematical challenges, games, and provocations.

From the inimitable (and inimitably punctuated) Gord! comes this collection of fourteen open-ended math puzzles. Each gives a kind of mechanism or guiding rule, which then lends itself to myriad (nay, infinite!) avenues of exploration.

Here’s the first (lightly paraphrased):

Write the numbers 1 to 25 in a 5-by-5 grid. Now drop a hungry bug on one of the squares. It eats that number, then moves up, down, left, or right to the lowest adjacent number, where it eats again, repeating this process until it can move no further.

Which starting numbers will allow the bug to traverse the whole grid?

(Example: if the bug starts on 14, it will not traverse the whole grid. It misses the bottom row.)

These “pickles” stretch the “low floor, high ceiling” concept to extremes. That is, they lead quickly into unsolved research problems (What about larger grids? What about rearranging the numbers? What if the grid is an arbitrary graph?), and also, they appeal to my five-year-old.

In other words: the floor is in Earth’s mantle, and the ceiling is in the stratosphere. Perfect for the math teacher in your life.

“Playful, eclectic, and ingenious,” is the unsolicited blurb I offered in an email; “in short, it’s unmistakably Walter.”

On second thought, I might call it “a museum of an irreplaceable mind.”

The book collects 239 emanations of what I call Joris Radiation. That includes two-player games, paper-folding activities, prompts for improvisation, and mathematical provocations (much in the spirit of Gord!’s pickles).

For example: If you draw a bunch of cross-shaped pentominos on graph paper, what’s the largest gap you can create, such that no cross can fit in the gap?

Or, for another: Write a string of digits. Then, remove any pair of digits, and replace them with the digits in their sum; for example, 9 and 9 can be replaced with 1 and 8. Repeat this process as long as you like. (Also, since zeros don’t affect sums, you may just leave them out.)

When is it possible to reduce the number string to nothing but 1’s?

It can take some work to decipher Walter’s instructions, and to distinguish the costume jewelry from the priceless gems. But if you want to swim in the stream of a bizarre and fabulous consciousness, you can do no better. Perfect for the creative weirdo in your life.

I’m not quite sure how to describe this book, so let me deploy one of the devices Bellos puts to great effect in the book: multiple choice. Is this book most appealingly described as…

1) A light, snackable compendium of puzzles requiring no special knowledge or expertise?

2) A curiosity-widening volume of puzzles drawn from various disciplines: probability, geography, logic, physics, game theory, and most delightfully, cognitive psychology?

3) A provocative collection of puzzles where your fiercest intuitions will betray you?

4) Another fabulous puzzle book from Alex Bellos, who is making a real habit of such things?

Over in his puzzle column at The Guardian, Alex has given some puzzles in the same style, and a couple from the book itself. Great for the Sudoku fan, the crossword fan, or just the fan of intellectual surprises and “ooh!” moments.

Not in the habit of buying books, I see? But you are in the habit of clicking hyperlinks? Well, I can’t endorse your lifestyle, but I can’t judge either; my own hyperlink-to-book ratio is functionally infinite.

Anyway, you’ll enjoy Tanya’s blog, which featured some classically elegant puzzles in 2024:

Great as those are, I was even more fond of Tanya’s open-ended “lateral thinking” puzzles (and the wonderfully varied answers Tanya’s students gave). In addition to heavy thingy and home late, here are three of my favorites:

Thrice makes sixty:

I can use the number 20 thrice to make 60: 20 + 20 + 20 = 60. Make 60 again by using a different number three times.

Clean sister, dirty sister:

One day, two sisters decided to clean the old shed at the bottom of their garden. When they finished cleaning, one had a dirty face and the other had a clean face. The sister with the clean face went and washed her face, but the girl with the dirty face did not wash. Why should this be so?

And my personal favorite, guessing the number in one question:

Mike thought of one of three numbers: 1, 2, or 3. He is allowed to answer “Yes”, “No”, or “I don’t know”. Can Pete guess the number in one question?

Puzzles with Bad Drawings

What about here, at this very blog? Well, I am pleased to highlight the lovely job my pal Chris did in creating the daily information-gathering puzzle L.A.P.

But also, this year’s posts included:

And I’ve started closing out my monthly roundup posts with a “parting puzzle.” They’re collected here (with spoilers). And while I’m at it, here’s one for this month, slightly different in flavor…

A Parting Puzzle: The Divided States of Economica

Fact: the U.S. has the world’s largest economy (as measured by GDP).

But let’s break the U.S. into smaller economies, and see how much of the world’s Top 10 it can thereby dominate.

(1) Warm-up: Treating the U.S. as 51 separate economies, one for each state (plus DC), how many rank among the world’s ten largest?

(2) Challenge: Now, consider all the ways to decompose the U.S. into new, smaller economies, each consisting of one or more contiguous states. Let’s do this to maximize the number of economies that rank in the world’s ten largest. What’s the best you can do?

Here’s a csv to get you started: