Category Archives: Puzzles

A Boggle Grid Big Enough For Every Word?

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Puzzly minds are often analytical minds, so I’m no stranger to wild attempts at puzzle-inspired calculation.

Puzzle people have previously calculated the human limit for solving a Rubik’s Cube, as well as the fastest known mechanical solve of a Rubik’s Cube. Years ago, I myself tried to figure out how many years it would take to use every unique Sudoku grid possible.

But optimization is something altogether different, and it’s a intriguing way to look at the potential of puzzles.

A Redditor by the name of AntiqueRevolution5 posed the following question a few days ago, and it absolutely fits the bill of both puzzly calculation and optimization:

What would a Boggle grid look like that contained every word in the English language?

Well, there are somewhere between 1 million and 1.5 million words in the English language, so I imagine it would be pretty big.

The redditor is an artist, and their goal is to make a sculpture composed of Boggle dice. And their concept is fascinating:

The idea for the piece is that it’s a linguistic Rorschach that conveys someone could find whatever they want in it. But it would be even cooler if it literally contained any word someone might reasonable want to say or write.

So, where do you start with something like this? Our artist has a suggestion:

To simplify, we could scope it to the 3000 most important words according to Oxford. True to the nature of Boggle, a cluster of letters could contain multiple words. For instance, a 2 x 2 grid of letter dice T-R-A-E could spell the words EAT, ATE, TEA, RATE, TEAR, ART, EAR, ARE, RAT, TAR, ERA. Depending on the location, adding an H would expand this to HEART, EARTH, HATE, HEAT, and THE...

What would be the process for figuring out the smallest configuration of Boggle dice that would let you spell those 3k words linked above? What if the grid doesn’t have to be a square but could be a rectangle of any size?

Naturally, creative minds accepted the challenge.

One user claimed that an online Boggle website called Squaredle has two 10×10 boards with just under 900 words of 4 or more letters.

Another user, a programmer, was able to create an 18×18 grid with approximately 450 words in under 10 minutes.

As you might expect, I was unable to resist diving into this one. But I’m not a programmer, so let’s do some meatball mathematics to get a sense of the scope of the puzzly conundrum before us.

We can extrapolate that if we get 450 words in an 18×18 grid, we’d need seven 18×18 grids connected to approach 3,000 words, assuming there’s some consistency in letter efficiency.

If we stick to the rectangular suggestion of the original post, a 63×36 rectangle (six 18×18 grids in a 2×3 arrangement, plus a 9×36 grid attached at the bottom) should allow for those 3,000 words.

Now, I can’t verify that. But 63×36 means 2,268 letters in the grid. Which, with a 3,000 word goal (including two- and three-letter words) kinda feels possible.

Of course, this is just to cover that 3,000 word list. Remember that the English language is estimated to contain between 1 million and 1.5 million words total.

That’s 333-and-a-third times more, if we use a million words. It’s 500 times more words if we assume 1.5 million words.

So, that’s 2,268 letters in our 3,000 word grid. Multiply that by 500 and you get 1,134,000 letters in the grid.

That means we’d need a grid that’s 1,065 x 1,065 to cover the entire English language.

So what does that mean in Boggle terms?

A standard six-sided die is 16 millimeters. That’s 17,040 millimeters, or 17.04 meters. That’s 67.09 inches. We’re talking about a Boggle game that’s FIVE AND A HALF FEET ACROSS.

That’s one heck of a Boggle grid.

Now, of course, these numbers are all estimates, and dubious ones at that. But I couldn’t resist TRYING to find an answer, even if it’s just a ballpark number.

You see, fellow puzzlers, this brand of puzzle efficiency tickles something in my brain, as there are several Penny Press puzzles I quite enjoy that employ a similar idea.

Starspell (pictured above) involves finding words in a star-shaped grid, except unlike Boggle, you can reuse letters. So you could bounce back and forth from A to N and spell BANANA, for instance.

Word Maze involves a small grid with many words hidden inside Boggle-style (though it’s actually a themed word list, meaning it’s not optimized to just cram as many words inside as possible).

Letter Perfect is a reversal of the idea, seeing if the solver can arrange letters in a mostly-empty 4×4 grid to fit every word in a given wordlist. It’s excellent training for a challenge like this, since you learn about efficiency of letter placement and how many words can spell out with neighboring letters if you’re clever.

I don’t know if any programmers will figure out how to build a language-spanning Boggle grid, but I look forward to seeing them try!

Happy puzzling, everyone.

What Makes a Good Brain Teaser?

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I was going through a collection of brain teasers a fellow puzzler gave me, and it occurs to me that “brain teaser” is one of the least specific puzzle terms around.

Riddles, logic and deduction, math puzzles, and wordplay games all fall under the brain teaser umbrella. So you never know what you’re gonna get. Are your math skills required? Your outside-the-box thinking? Your ability to pay attention to the specifics of the question itself?

In this collection alone, I found examples of each of these types of puzzles:

Riddle: Sometimes I’m green, sometimes I’m black. When I’m yellow, I’m a very nice fellow. That’s when I’m feeling mighty a-peeling. What am I?

Logic / deduction: 3 guys go into a hardware store, all looking for the same thing. William buys 1 for $1. Billy buys 99 for $2. Finally, Willie buys 757 for $3. What were they buying?

Math puzzle: The sum is 12 and the same digit is used 3 times to create the sum. Since the digit is not 4, what is the digit?

Wordplay: Which state capitals would you visit to find a ram, cord, bus, and dove?

So, if someone challenged you to a brain teaser, these would all be fair game. Would you be able to solve all four of them?

It’s possible you wouldn’t, because good and bad brain teasers alike employ tricks to keep you on your toes.

Some hide the answer in plain sight:

Homer’s mother has 4 children. 3 of them are named Spring, Summer, and Autumn. What is the 4th named?

Some use misdirection, purposely phrasing the question to get you thinking one way and steering you away from the real solution:

The big man in the butcher shop is exactly 6’4″ tall. What does he weigh?

In fact, both of these examples use plain sight (Homer, butcher shop) and misdirection (implying a pattern with seasons, specifying his height) to distract you.

I suspect you weren’t fooled by either of them, though.

Others try to overwhelm you with information so you bog yourself down in the details instead of clearly analyzing the problem at hand:

Nina and Lydia start from their home and each runs 2 miles. Nina can run a mile in 8 minutes 30 seconds and Lydia can run a mile in 9 minutes 10 seconds. When they finish running, what is the furthest apart they can be?

A lot of numbers get thrown at you, but they’re irrelevant, since the question only asks about the distance, not the time. So if they each run 2 miles, the furthest apart they can be is 4 miles. The rest is just smoke.

These are all effective techniques for teasing a solver’s brain. You’re given all the information you need to solve the puzzle, plus a little extra to distract, mislead, or overwhelm you.

Unfortunately, some brain teasers use unfair techniques to try to stump you:

Jacob and Seth were camping in June. Before going to sleep they decided to read a book. They both agreed to stop reading when it got dark. They were not fast readers, but they finished the entire encyclopedia. How?

Ignoring the fact that these two boys somehow brought an entire encyclopedia with them on a camping trip, we’re not actually given a lot of information here.

So that makes the intended answer seem like more of an insane leap than a logical jump to the conclusion: They were in Lapland, land of the midnight sun, and the sun didn’t set until September.

WHAT?

There’s no reasonable way for someone to reach this conclusion based on the information given. In fact, it makes less sense the more you read it. Presumably Jacob and Seth know where they are camping, and that it wouldn’t get dark for months. So why would Jacob and Seth agree to stop reading when it got dark IF THAT MEANT THEY’D BE READING UNTIL SEPTEMBER!?

This is gibberish, and you’d be surprised how often something like this gets passed off as acceptable in a collection of brain teasers. (I discussed a similar issue with detective riddles in a previous blog post.)

Let’s close out today’s discussion of the ins and outs of brain teasers with a few fun, fair examples, shall we?

  • In what northern hemisphere city can you find indigenous tigers and lions?
  • Scientists have found that cats are furrier on one side than the other. The side with the most fur is the side that cats most often lie on. Which side of a cat has more fur?
  • A woman married over 50 men without ever getting divorced. None of the men died and no one thought she acted improperly. Why?

Did you solve them all? Let us know! Also, please share your favorite brain teasers (or your tales of treacherous and unfair brain teasers) in the comments below!

Happy puzzling, everyone!

A Surprise Escape Room… During a Wedding!

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We’ve seen puzzly marriage proposals (even helping design a few over the years!), we’ve seen puzzly wedding party invitations, and we’ve seen puzzly wedding receptions…

But I think this is the first puzzly wedding I’ve ever seen!

Yes, the bride surprised the groom with a wedding ring trapped in a combination lock!

She then read out a riddle, loaded with clues that were tied to costumes and items held by various guests in attendance, each of them with a number.

So he needed to decode which references in the riddle connected to the numbers and characters in the audience.

And, quite cleverly, the combination turned out to be the exact date they met.

This is the lovely sort of puzzle fun that I think all of us sorely need right now, so thank you to friend of the blog and all around good egg Jen for sharing the video with me!

And if you’re in the mood for other lighthearted, puzzle-fueled, wedding-centric content, here’s a quick bullet list of links for you to enjoy:

Happy puzzling, everyone! Happy Valentine’s Day! And watch out for those escape rooms… they can show up anywhere these days!

puzzlelove

Constructors, What’s Your Salomon Number?

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I’m a nerd.

I know that probably comes as a huge shock to you, fellow puzzler. Positively astonishing that a guy who has spent over twenty years making puzzles — more than ten writing about them here — and even more years running D&D games, LARPs, escape rooms, and murder mystery dinners is a nerd.

But that’s the truth.

And as a nerd, I love watching people quantify things in strange ways. Weird units of measurement like smoots, or how FEMA uses whether local Waffle Houses are open to determine the severity of natural disasters, or the Muta Scale used by wrestling fans to determine how bloody a wrestling bout is.

So when a recent Puzzmo crossword namedropped the Salomon number, I immediately wanted to know more.

Created by crossword constructor and super cool website owner Quiara Vasquez, the Salomon Number is named in honor of Nancy Salomon, a prolific and influential crossword constructor whose generosity, mentorship, and creativity still shapes the world of crosswords today.

Akin to the Bacon Number (how many steps it takes to link someone to Kevin Bacon through shared performances) and the Erdos Number (how many steps it takes to link someone to Paul Erdos through co-authored papers), the Salomon Number represents how many steps it takes to link someone to Nancy Salomon through collaborative puzzling.

Nancy Salomon collaborated with over forty constructors over the years, so there’s a very healthy talent pool to build a Salomon Number through.

You’re welcome to play the game mentally, but if you need a helping hand, Alex Boisvert has created an automated system for generating Salomon Numbers!

A visual graph of the Salomon Number network.

I tested it by searching for several of my very favorite puzzle people.

I was pleasantly surprised to discover that Los Angeles Times crossword editor and puzzle badass Patti Varol had a Salomon Number of 2, thanks to a collaboration with Matt Skoczen.

Crossword gentleman Doug Peterson had a Salomon Number of 3, linked by Joon Pahk and Brendan Emmett Quigley.

As I explored the database, it quickly became apparent how small a world crosswords can be — despite all the wonderful new voices entering the field over the last few years — so I started to wonder…

What’s the largest Salomon Number in the system?

I started with the celebrity constructors during the 75th anniversary celebration of The New York Times a few years ago.

Weird Al Yankovic’s number was 4. A good start. But this quickly petered out, as many of those celebrities were paired with veteran constructors with strong Salomon Numbers.

I pored over the list for familiar names to see their scores. Gaby Weidling’s number was 5, which was the highest I’d seen so far.

But I was surprised to find some prolific cruciverbalists in the database with no Salomon Number at all. With so many indie crossword outlets, there’s a chance there are collaborations that aren’t included in the current database, so there’s always the possibility of a successful search later.

(I also couldn’t resist doing a little sleuthing myself to see if I could uncover connections that weren’t in the database. Nothing yet, but I definitely can’t resist more digging later.)

But as constructors collaborate and innovate going forward, it’s going to be fascinating to watch this network continue to expand and complicate.

Make sure to check out Quiara’s post, which dives deeper into Salomon Numbers and even ponders Bacon-Erdos-Salomon Numbers!

Happy puzzling, everyone!

A New Medium for Optical Illusions: Quilting?

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Optical illusions are puzzles for the eye, a visual treat that tricks you into seeing things that aren’t there. These inspired bits of perceptual trickery can fool you into thinking near is far, big is small, or two dimensions are really three.

The advent of computers has helped push the boundaries of optical illusions, with eye-catching tessellations and visual effects that confuse the eye into thinking they see moving objects or impossible figures.

This makes it all the more impressive when people can accomplish the same in a purely physical medium.

Like quilting.

People accomplish some amazing designs with quilting, like this labyrinth quilt:

The dimension in this one is fantastic. (Though the puzzle nerd in me must point out that it’s not exactly solvable. Even to a minotaur.)

And then there’s this eye-popping wonder:

I can only imagine the amount of work required to get all of those squares cut and positioned the right way to create that bulging illusion in the middle of the checkerboard pattern. Talk about ambitious!

Quilts like those above are incredible efforts. But they feel doable. They feel natural when you look at them, even as they wow you with their exacting detail.

But nothing has amazed me quite as thoroughly as this glitch quilt pattern designed by Modern Groove Quilts.

The precision and color play required to create this illusion is a mathematical wonder.

It’s literally hard to look at! My eyes keep protectively sliding away from the design just to prevent my brain from struggling with it.

It feels like my computer screen is malfunctioning.

Using a rug as a backdrop is one thing, but in the wild, where nature is crisp and clear and the quilt appears blurry, it’s a baffling visual experience.

The only thing I can liken it to? This tattoo that looks blurred but isn’t:

These artistic, handmade optical illusions really show that, computer-assisted or not, it’s the ever-evolving ingenuity of creators that keeps the world full of wonder and creative achievement.

Now everyone, go stare at something boring and rest your eyes for a while.

(Oh, and check out the other patterns available from Modern Groove Quilts!)

Happy puzzling, everyone!

Minotaurs Don’t Make Sense?

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When you think of mazes, there’s probably no figure more iconic than the legendary minotaur.

The Bull of Minos — the direct translation of “minotaur” — called an elaborate labyrinth on the island of Crete home. This labyrinth, built by Daedalus, was said to be inescapable.

Even if you’re not a fan of Greek mythology, you probably know the basics of the story. Unsolvable maze. Half-man, half-bull. It’s all pretty simple.

So you can imagine my surprise when I stumbled upon not one, but TWO different videos talking smack about one of my favorite mythical creatures!

The first one discussed the intelligence of the minotaur:

Now, a very simple misconception lay at the heart of this video. The speaker is conflating two different definitions of labyrinth.

The first is the traditional idea of an elaborate maze full of intricate passageways, blind alleys, and torturous complexity.

The second refers to labyrinth gardens, which are designed to be walked slowly, meditatively, and take you on a journey through a series of paths and concentric circles, but one that can be easily navigated or escaped. None of the trickery or challenge of a traditional labyrinth.

So when the above video claims that a minotaur would have to be pretty stupid to fail to escape the labyrinth, they’ve got the wrong labyrinth in mind.

This brings us to the second video of the day, which discusses how minotaurs have been adapted for the Dungeons & Dragons roleplaying game:

The speaker claims that D&D’s minotaurs don’t make sense, since they “can never get lost in a maze and will always find a way out.”

So, I went to verify what he said.

He referred specifically to the 5th edition version of the minotaur, so I checked both the 5th edition Monster Manual and the Monsters of the Multiverse sourcebook released later.

In the MM, all I find is this: “the minotaur can perfectly recall any path it has traveled.”

So what? There are a LOT of paths in a labyrinth, and remembering which ones you’ve walked doesn’t necessarily make you better at finding your way out.

Plus, how long is a path? How is the start or end of a path defined? Maybe our life is one LONNNNNG path, and he’s gotta recall ALL of it?!

Image courtesy of Wikipedia.

This ability is restated in the Monsters of the Multiverse book as “you always know which direction is north”. That’s it. Cool. That’s great for navigation, but also not terribly helpful when it comes to solving mazes.

So I’m not really sure where he’s getting the whole “effortlessly defeats mazes” thing. I mean, the original Minotaur never bested the labyrinth. He was killed by Theseus there.

But, for the sake of thoroughness, I grabbed my 3rd edition Monster Manual off the shelf to see what it had to say about minotaurs.

The book states that minotaurs can’t ever become lost and are “immune to maze spells” (which banish someone to an extradimensional labyrinth).

“Can’t ever become lost” is incredibly open to interpretation. Does it always know the way out of a place? Or does it always know WHERE it is in a vague sense? There’s a big difference between knowing where the exit of a labyrinth is and simply knowing you’re in a labyrinth.

Like, if I know I’m in New York City, does that mean I have any idea where I’m going? Not necessarily.

Image courtesy of Do You Maze.

So, where does this leave us?

Well, in short, it tells me that people don’t necessarily know what a labyrinth is or what it means to have the skills of a minotaur.

After all, do we know if the original Minotaur ever really wanted to leave? He got regular deliveries of food, and was otherwise left to his own devices.

Doesn’t sound so bad, really. Anyone know a labyrinth that’s hiring right now?