Two Brain Teasers, Courtesy of Conway

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Last week, we penned a post celebrating the life and puzzly legacy of mathematician John Horton Conway, and several of our fellow PuzzleNationers reached out with their own thoughts or questions about Conway.

One recurring subject was about his love of puzzles and what kind of puzzles he enjoyed solving. So, naturally, I went hunting for some of Conway’s favorite puzzles.

As it turns out, Alex Bellos of The Guardian had me covered. Alex has a recurring puzzle feature on The Guardian‘s website where brain teasers and other mental trickery awaits intrepid solvers.

Years ago, Alex had asked Conway for suggestions for his column, and Conway offered up two tricky puzzles.

And now, I happily share them with you.


#1: The Miracle Builders

I had a window in the north wall of my house. It was a perfect square, 1 meter wide and 1 meter high. But this window never let in enough light. So I hired this firm, the Miracle Builders, who performed the impossible. They remodeled the window so it let in more light. When when they’d finished the window was a perfect square, 1 meter high and 1 meter wide.

How did they do it?


#2: The Ten Divisibilities

I have a ten digit number, abcdefghij. Each of the digits is different.

The following is also true:

  • a is divisible by 1
  • ab is divisible by 2
  • abc is divisible by 3
  • abcd is divisible by 4
  • abcde is divisible by 5
  • abcdef is divisible by 6
  • abcdefg is divisible by 7
  • abcdefgh is divisible by 8
  • abcdefghi is divisible by 9
  • abcdefghij is divisible by 10

What’s my number?

[To clarify: a, b, c, d, e, f, g, h, i, and j are all single digits. Each digit from 0 to 9 is represented by exactly one letter. The number abcdefghij is a ten-digit number whose first digit is a, second digit is b, and so on. It does not mean that you multiply a x b x c x…]


Did you solve one or both of these fiendish mind ticklers? Let us know in the comments section below! We’d love to hear from you.

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Celebrating the Puzzly Legacy of John Horton Conway

The worlds of puzzles and mathematics overlap more than you might think. I’m not just talking about word problems or mathy brain teasers like the Birthday Puzzle or the jugs of water trap from Die Hard with a Vengeance.

For twenty-five years, Martin Gardner penned a column in Scientific American called Mathematical Games, adding a marvelous sense of puzzly spirit and whimsy to the field of mathematics, exploring everything from the works of M.C. Escher to visual puzzles like the mobius strip and tangrams. He was also a champion of recreational math, the concept that there are inherently fun and entertaining ways to do math, not just homework, analysis, and number crunching.

And on more than one occasion, Gardner turned to the genius and innovative thinking of John Horton Conway for inspiration.

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[Image courtesy of Wikipedia.]

Conway was best known as a mathematician, but that one word fails to encapsulate either his creativity or the depth of his devotion to the field. Conway was a pioneer, contributing to some mathematical fields (geometry and number theory among them), vastly expanding what could be accomplished in other fields (particularly game theory), and even creating new fields (like cellular automata).

Professor of Mathematics, Emeritus, Simon Kochen said, “He was like a butterfly going from one thing to another, always with magical qualities to the results.” The Guardian described him in equally glowing terms as “a cross between Archimedes, Mick Jagger and Salvador Dalí.”

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[Image courtesy of Cornell.edu.]

His most famous creation is The Game of Life, a model that not only visually details how algorithms work, but explores how cells and biological forms evolve and interact.

Essentially, imagine a sheet of graph paper. In The Game of Life, you choose a starting scenario, then watch the game proceed according to certain rules:

  • Any live cell with fewer than two live neighbors dies, as if by underpopulation.
  • Any live cell with two or three live neighbors lives on to the next generation.
  • Any live cell with more than three live neighbors dies, as if by overpopulation.
  • Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

The process plays out from your starting point completely without your intervention, spiraling and expanding outward.

It’s the ultimate if-then sequence that can proceed unhindered for generations. It is a literal launchpad for various potential futures based on a single choice. It’s mind-bending and simple all at once. (And you can try it yourself here!)

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[Image courtesy of Sign-Up.To.]

But that’s far from Conway’s only contribution to the world of puzzles.

Not only did he analyze and explore puzzles like the Soma cube and Peg Solitaire, but he created or had a hand in creating numerous other puzzles that expanded upon mathematical concepts.

I could delve into creations like Hackenbush, the Angel Problem, Phutball/Philosopher’s Football, Conway’s Soldiers, and more — and perhaps I will in the future — but I’d like to focus on one of his most charming contributions: Sprouts.

Sprouts is a pencil-and-paper strategy game where players try to keep the game going by drawing a line between two dots on the paper and adding a new dot somewhere along that line.

The rules are simple, but the gameplay can quickly become tricky:

  • The line may be straight or curved, but must not touch or cross itself or any other line.
  • The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.
  • No spot may have more than three lines attached to it.

Check out this sample game:

sprouts

[Image courtesy of Fun Mines.]

It’s a perfect example of the playfulness Conway brought to the mathematical field and teaching. The game is strategic, easy to learn, difficult to master, and encourages repeated engagement.

In a piece about Conway, Princeton professor Manjul Bhargava said, “I learned very quickly that playing games and working on mathematics were closely intertwined activities for him, if not actually the same activity.”

He would carry all sorts of bits and bobs that would assist him in explaining different concepts. Dice, ropes, decks of cards, a Slinky… any number of random objects were mentioned as potential teaching tools.

Professor Joseph Kohn shared a story about Conway’s enthusiasm for teaching and impressive span of knowledge. Apparently, Conway was on his way to a large public lecture. En route, he asked his companions what topic he should cover. Imagine promising to do a lecture with no preparation at all, and deciding on the way what it would be about.

Naturally, after choosing a topic in the car, the lecture went off without a hitch. He improvised the entire thing.

Of course, you would expect nothing less from a man who could recite pi from memory to more than 1100 digits? Or who, at a moment’s notice, could calculate the day of the week for any given date (employing a technique he called his Doomsday algorithm).


Conway unfortunately passed away earlier this month, due to complications from COVID-19, at the age of 82.

His contributions to the worlds of mathematics and puzzles, not to mention his tireless support of recreational math, cannot be overstated. His work and his play will not soon be forgotten.

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[Image courtesy of Macleans.]

If you’d like to learn more about Conway, be sure to check out Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts.

[My many thanks to friend of the blog Andrew Haynes for suggesting today’s subject and contributing notes and sources.]


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There’s a Little Something Extra in These X-Words…

Crossword constructors can be fiendishly clever, so there’s often something extra lurking inside a crossword grid, if you know where to look.

Sometimes it’s easy to spot. There are shaded areas or circled letters to reveal the hidden bonus answers that add a touch of pizzazz to a grid.

For instance, our friends at Penny/Dell Puzzles have a recurring crossword variant, Revelation, which conceals a quotation in a standard crossword grid.

The New York Times crossword has also featured this gimmick in puzzles plenty of times, perhaps most notably in a May 2015 puzzle where both poet WILLIAM CARLOS WILLIAMS and the title of his poem THE LOCUST TREE IN FLOWER read down the sides of the grid, and the circled letters within the grid concealed the poem in full!

[Image sourced from Amy Reynaldo’s Diary of a Crossword Fiend.]

For his puzzle featured in an episode of The Simpsons, constructor Merl Reagle famously snuck a message into another New York Times crossword puzzle, allowing Homer to apologize to Lisa for his transgressions in the most public puzzly forum possible.

If you went diagonally from the upper left to the lower right of the grid, the statement “Dumb dad sorry for his bet” could be found.

[Image courtesy of The Guardian.]

Whether it’s a hidden quotation or a secret message hiding amidst the black squares and crisscrossing entries, these bonus answers offer a final little twist that wow solvers, leaving them shaking their heads at the cleverness and skill of constructors.

A puzzle in The Wall Street Journal recently reminded me of another surprise that a crafty constructor can spring on an unsuspecting solver.

This particular puzzle from September 28th of this year had instructions instead of the usual themed answers. If you read 22 Across, 61 Across, and 105 Across, you received the following message: Find the names of ten gems / hidden within the puzzle / grid in word search style.

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[Image courtesy of Reddit.]

Yes, the appropriately titled “Treasure Hunt” by Mike Shenk had jewels hidden among the answers in the grid, reading horizontally, vertically, and diagonally, just as they would in a word seek or word search.

Although this led to a few awkward entries — GOT ENRAGED is a bit clunky for an answer, even if the goal is to hide GARNET backwards within it — the grid is mostly great, and the spread of gems — from DIAMOND and EMERALD to ONYX and TOPAZ — is impressive. (I particularly liked RUBY reading out backwards in HURLYBURLY.)

I haven’t encountered many of these word search-style crossword surprises over the years, but there is one other prominent example that came to mind.

In his second appearance in today’s post, Merl Reagle constructed a special puzzle to celebrate the 100th anniversary of the crossword in 2013.

His puzzle was converted into a solvable Google Doodle — you can still solve it here! — and Merl added a crafty word search element by hiding the word FUN multiple times in the grid.

Why “fun,” you ask? Because that was the set word in Arthur Wynne’s original “word-cross” puzzle over one hundred years ago.

Believe me, constructing a great crossword grid is taxing enough. Adding touches and tricks like these just ratchet up both the difficulty involved and the skill level required to make the whole endeavor a harmonious success.

Kudos to those, past and present, who have pulled it off with such style.


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A Clued Sudoku Puzzle? It’s More Than Meets the Eye

cryptic

The hunt is always on for the next big puzzle idea.

Sometimes, it’s an old idea that gets repackaged and catches fire. That’s what happened with Sudoku, a puzzle that had been around since the late ’70s, but only rose to prominence decades later.

Other times, it’s a combination of different puzzle types that yields something special. Our friends at Penny Dell Puzzles, for instance, have a popular puzzle called Anagram Magic Square, which combines crossword-style cluing, anagrams, and the mathematical element of a magic square to create an engaging puzzle experience you can solve from several angles.

Whether a puzzle is destined for superstardom or not depends on a lot of factors: difficulty, the type of solving it involves, how intuitive the solving is (i.e. needing a lengthy explanation vs. getting the gist of the puzzle from a glance), visual aesthetics, and more.

As a puzzler, it’s always exciting to try out a new puzzle. Wholly original ideas are rare, to be sure, but even a single twist on an old classic can be enjoyable if executed well.

Today, we’re taking a look at a puzzle that combines Sudoku with cryptic crosswords (aka British-style crosswords). It’s called Cluedoku, and it was created by cryptic constructor Charlie Methven, better known in solving circles as Chameleon, a contributor to British puzzle outlets like The Guardian.

cluedoku

[Just a sample of the puzzle. Check out the entire puzzle here.]

Like Sudoku, Cluedoku involves placing the digits 1 through 9 into each row, column, and 3×3 square in the grid. But unlike Sudoku, there are no set letters.

Instead, you have 81 clues, one for every cell in the grid, utilizing cryptic-style cluing to hint toward which of the nine numbers goes in a given cell.

Once you’ve unraveled a clue and placed a number in the grid, standard Sudoku rules apply: that number will only appear once in a row, column, or 3×3 square.

But that’s easier said than done. These clues run the gamut of slyly clever to almost baffling. Even when you consider that there are only nine possible answers for each clue, it’s still a challenge. (Plus, not all of the clues adhere to the standard cryptic cluing mechanic of having both a definition AND a wordplay clue included.)

That being said, you’ll find lots of traditional cryptic cluing tricks at play here.

Now, we’re going to be discussing specific clues and answers from this puzzle, so this is your spoiler warning.

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Last chance to solve without spoilers!

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Okay, here we go.

In terms of traditional cryptic cluing gimmickry, we see hidden words, anagrams, homophones, word reversals, and more.

In clue 6 — Axis revolves without beginning to accelerate — we revolve (aka reverse) axis to spell SIXA, and then drop the A (“without beginning to accelerate”) to spell SIX.

In clue 8 — Prime cut from sloth reeks — the answer hides in plain sight, as a prime number (three) reads out in sloTH REEks (and can be cut out of it).

In clue 22 — Scenes in X-Men Origins reveal how many claws Wolverine has! — the phrase “origins reveal” points towards the first letters of the words that precede it proving the answer, meaning that SIX is the number of claws Wolverine has (three on each hand).

There is a similar game in clue 67 — With only seconds remaining, Officer Columbo outwits crook — which has the second digits of “Officer Columbo outwits crook” spelling out FOUR.

In clue 27 — UFO demolished third of Parliament Square — the letter R (“third of Parliament”) gets mixed up with UFO to make FOUR, a square.

crossword1

But other clues would be familiar to crossword solvers in America.

Clue 29 — Number of Romans in the New Testament? — is simple wordplay for 6, since Romans is the SIXth book. (Similarly, clue 62 — Number of lines taken by bar staff — is a reference to the FIVE lines that make up a staff in sheet music.)

Clue 34 — Top score in Scrabble — is a bit more devious, requiring you to know that T is worth 1 point, O is worth 1 point, and P is worth 3 points, making the correct answer FIVE.

Clue 48 — Man’s arms’ legs’ digit — feels like a clue you’d see at the Indie 500 or Lollapuzzoola, because it’s initially baffling, but then reveals itself as merely clever and challenging. You see, there are THREE legs on the coat of arms for the Isle of Man. But that’s concealed by the wordplay involving three different words that don’t mean what you’d think.

This mix of American and British-style clues made for a fun solve that mixed and mingled two worlds of cluing nicely.

I think my favorite clue was Clue 39 — 192+284 — because it was built like one of those magazine word puzzles, the ones where “rockcaughthardplace” means “caught between a rock and a hard place.” In this case, you have “2+2” literally in 1984. And for anyone familiar with George Orwell’s famous novel, 2+2 in 1984 equalled FIVE.

Although obviously Cluedoku isn’t really sustainable as a recurring puzzle — you’d burn out your anagrams and homophones pretty quickly, as Chameleon himself stated in an interview — it is an impressive marriage of two different puzzles that rarely interact otherwise.

But he did raise the possibility of another variation in the future:

If I did another Chameleon cluedoku, I think I’d use the seven colours of the rainbow plus black and white, as solvers could then colour in each square as they solved. How’s “Cry over Norwich’s core Canary”?

That sounds like a fun follow-up to an interesting puzzle.

What did you think of Cluedoku, fellow puzzlers and PuzzleNationers? Let us know in the comments below! We’d love to hear from you.


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How to Get Started in Cryptic Crosswords

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[Image courtesy of Amazon. The Times Cryptic Crossword Book.]

On Twitter yesterday, Oliver Roeder from FiveThirtyEight asked, “If one wanted to learn/practice solving cryptic crosswords, with what puzzles should one begin?”

Most of the replies mentioned different cryptic crossword outlets to try out, like The Nation, Harper’s Magazine, and The UK Times Quick Cryptic Crossword Book. The Nation in particular was recommended as a good starter cryptic.

Monthly offerings from constructors like Andrew Ries and Cox & Rathvon were also mentioned, though I would add Patrick Berry’s Son of the Crypt cryptic collection to the list of suggestions. (I would normally also recommend The Guardian because of their great cryptics, but they’re pretty tough, particularly for beginners.)

This, of course, presumes that Roeder meant which cryptic puzzles one should start with.

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[Image courtesy of The New European.]

It occurred to me that he might be asking what OTHER puzzles are good for beefing up your cryptic crossword solving game.

Given the different kinds of clues used in cryptic crosswords, I have a few suggestions.

1. Anagram puzzle

Anagrams are a staple of cryptic cluing, and any puzzler looking to get into cryptics should have some facility with them. There are plenty of ways to practice — the Jumble, Anagram Magic Square and other puzzles from our friends at Penny Dell Puzzles, and even Bananagrams, Words with Friends, or Scrabble will help build your anagram skills.

2. Rebus

Rebus puzzles are all about adding and subtracting letters to form words or phrases, and there’s plenty of that in cryptic cluing. This is a good way to get used to breaking down longer words into abbreviations, anagrams, and so on in order to puzzle out the answer to a cryptic clue.

3. Brain teaser/riddle

Many cryptic clues rely on words with multiple meanings, as well as words that serve as both instructions and hints. Brain teasers and riddles employ similar wordplay, and they can help you develop a proclivity for looking at words from a new point of view.

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[Image courtesy of Eastern Daily Press.]

Of course, if you want help learning to decipher the many variations on cryptic crossword cluing that you’ll encounter, there are some great resources out there.

Penny Dell Puzzles has a PDF containing examples of the most common cluing tricks, and you can bolster that with similar insights from Wikipedia and The Nation.

If you’re looking for deeper dives into all sorts of cryptic cluing, my one-stop shop for insight is The Guardian’s crossword blog. They offer regular features breaking down various kinds of cryptic clues.

In the last few weeks alone, they’ve covered cycling clues, “stuttering” in clues, and how the points of the compass can be used in cluing, and there are dozens of similarly illuminating posts in their archive.

It’s a terrific resource for newbie cryptic solvers and established puzzlers alike.

And it’s worth getting into cryptic crosswords, if only for the occasional subversive little Easter egg like this one from yesterday’s The Guardian cryptic:

brexit

Did I miss any resources or outlets for great cryptic crosswords? Let me know in the comments section below! I’d love to hear from you!


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Getting Started with Crosswords

We spend a lot of time talking about crosswords here on PuzzleNation Blog, and rightfully so.

For more than a century now, crosswords have been the standard-bearer for paper-and-pencil puzzles. From your local paper to The New York Times crossword, from online solving to puzzle apps like our very own Penny Dell Crosswords App, crosswords sit comfortably at the apex of the proverbial puzzle mountain, atop worthy also-rans like word searches, cryptograms, and Sudoku.

[Apparently Puzzle Mountain is actually a place. Who knew?]

But in talking about crosswords, it’s easy to forget that not everyone solves them. In fact, plenty of people find them intimidating, given the mix of trivia, wordplay, and tricky cluing that typify many crosswords these days, particularly in outlets like The New York Times, The LA Times, The Guardian, and more.

So today, I thought I’d offer some helpful resources to solvers just getting started with crosswords.

First off, if you need help filling in troublesome letter patterns, Onelook is an excellent resource. Not only can you search for words that fit various patterns, but you can narrow your searches according to cluing, look up definitions and synonyms, and even hunt down phrases and partial phrases.

Along the same lines, there are websites like Crossword Tracker that offer informal cluing help culled from online databases. For something more formal, there’s XWordInfo, an online database of entries and cluing that also serves as an archive of NYT puzzles you can search for a small fee.

The NYT Wordplay Blog chronicles each day’s puzzle, including insights into the theme, key entries, and more, plus they’ve begun amassing helpful articles about crossword solving. Not only are there sample puzzles to download and solve to get you started, but there are lists of opera terms, rivers, and sports names to know to make you a stronger solver.

And if British-style or cryptic crosswords are your puzzle of choice, look no further than The Guardian‘s Crossword Blog, which frequently posts about various cluing tricks employed by crafting cryptic puzzle setters. Their “Cryptic Crosswords for Beginners” series of posts has discussed all sorts of linguistic trickery, covering everything from the NATO alphabet to elementary chemistry.

For other variety puzzles, our friends at Penny Dell Puzzles offer sample puzzles and helpful solving tips for many of the puzzles in their magazines. For example, you can find a sample Kakuro or Cross Sums puzzle on the page for their Dell Collector’s Series Cross Sums puzzle book, as well as a How to Solve PDF.

Is there a particular puzzle that troubles you, or one you find too intimidating to tackle, fellow puzzlers? If so, let us know! We can either point you toward a solving resource or tackle the puzzle ourselves in a future post to provide helpful solving tips!


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