Author Archives: gmdirect

Sudoku: Every number in its place.

Posted on by
Reply

Sudoku puzzles are as ubiquitous as reality shows these days, and enthusiastic solvers can find puzzles of nearly any difficulty with ease.

Sudoku puzzles are usually ranked from one to five stars, with five star puzzles being the most difficult. Difficulty can depend not only on the number of starting digits, but their placement and the level of deduction involved.

Sudoku enthusiasts were the first to notice that the lowest number of clues required for a unique solution is 17. Puzzles with 16 clues invariably had alternate solutions.

For comparison purposes, the average newspaper sudoku has 25 set numbers. The sudoku puzzles on PuzzleNation vary in difficulty, but our easy puzzles range from 30 to 36 clues and our expert puzzles range from 25 to 30 clues, with medium and hard puzzles clue counts falling in between.

But the 17-clue threshold was all conjecture until a mathematician from University College Dublin named Gary McGuire put in the time (and the computer processing power) to write a mathematical proof confirming the suspicions of sudoku enthusiasts.

He designed a specific computer algorithm to process various sudoku grid patterns, allowing him to cut down on the computing time necessary to verify his conjecture. (Even with the reduced computing time, it took 7 million CPU hours in total, a monumental amount of processing time.)

From the nature.com article:

The idea behind this was to search for what he calls unavoidable sets, or arrangements of numbers within the completed puzzle that are interchangeable and so could result in multiple solutions. To prevent the unavoidable sets from causing multiple solutions, the clues must overlap, or ‘hit’, the unavoidable sets. Once the unavoidable sets are found, it is a much smaller—although still non-trivial—computing task to show that no 16-clue puzzle can hit them all.

Of course, as I said before, difficulty isn’t just about the number of clues. The puzzle widely regarded as the world’s hardest sudoku puzzle has 21 clues, but their placement makes for a much more mentally-taxing solving exercise.

I guess it just goes to show the old real estate cliche is true: it’s all about location, location, location.

Puzzles for the Eye

Posted on by
4

I’m a huge fan of optical illusions and visual trickery. From trompe l’oeil paintings to the works of M.C. Escher, these pieces are sources of wonder, employing forced perspective and visual sleight-of-hand to create impossible objects and images with unexpected depth.

Essentially, they’re visual puzzles, left for you to sort out and examine at your leisure.

One of my favorites is also one of the simplest examples of multiple-perspective imagery: the Necker Cube.

As you can see, the Necker Cube appears to be in different configurations, depending on which part of the cube your eye interprets as facing you. By focusing on a different spot, the perspective shifts, and suddenly the cube is positioned differently.

There are numerous mindbending variations of the Necker Cube, some drawn as impossible figures, and others expanding on the illusion to further engage and disorient the viewer. Check this one out:

My eyes twitch a little just looking at it.

Usually, Necker Cube-style illusions have only two options, two perspectives between which they can shift. (Like in the old woman/young woman image that kicks off this post.)

But I recently discovered the following video, which features a three-dimensional image that can be viewed from three different perspectives:

It’s fascinating stuff, a perfect reminder that puzzles can be wordless and lurking in plain sight.

Check this out, mate.

Posted on by
4

DIY puzzling is only limited by your imagination and inventiveness. After all, pencil and paper is all you need for most improvised puzzle games (hangman comes to mind, for instance).

But if you require a bit more inspiration for some down-home puzzling, look no further than a chess or checkerboard.

I’ve written about knight’s tours before, but that’s just the tip of the iceberg when it comes to chess-based puzzles and brain-teasers.

You could challenge yourself with an Eight Queens puzzle. It sounds easy but it’s deceptively difficult. Place eight queens on a chessboard so that no queen is attacking (or capable of attacking) another.

This means that no queen shares a row, column, or diagonal with another queen.

There are only 12 distinct solutions, though each one can be rotated 90, 180, and 270 degrees, as well as mirrored on the board, leading to a much larger number of acceptable solutions. Numerous variations on this theme are available to test your wits and spacial reasoning, including placing 32 knights on the board without conflict, or nine queens and a pawn.

If that’s not your cup of tea, and you’ve got paper and scissors handy, you could whip up a quick game of pentominoes.

A pentominoes game consists of placing all of the above pieces into a given space without overlapping. You can play this on your own as a mental challenge, or add a bit of competition and strategy to it by alternating turns with other players and seeing who is the last person to place a piece on the board.

While you can use any size square or rectangle for your game board, a chessboard works well, especially since you can fit every piece on the board (leaving only a 2×2 space of uncovered squares). By setting or moving around this 2×2 space, you’ve instantly created a new challenge for yourself.

Personally, I’ve found them to be excellent palate cleansers after a few mentally-exhausting rounds of chess (though there are always variants on the game itself, like Pardu Ponnapalli’s TrimChess (image 2) and Jose Raul Capablanca’s expanded chess, if you’re looking for fresh ways to play.)

In any case, these are wonderful challenges with a minimum of set-up time, perfect for puzzlers looking for a new challenge without a lot of fuss. I hope you enjoy.

Clever cluing and outlaw wordplay…

Posted on by
5

Last week I collected some of my favorite examples of clever crossword cluing, hoping to highlight some marvelous wordplay along the way.

To do so, I called in the cavalry and recruited the help of a half-dozen or so fellow puzzlers. As the terrific, funny, and cunning clues poured in, some of them were accompanied by notes, indicating that one publisher or another had barred them from publication.

Some clues, no matter how funny or clever, are rejected by publishers who view them as inappropriate for public consumption.

I respectfully disagree. =)

Heck, I wrote an entire blog post about cluing last year that featured my favorite not-quite-appropriate-for-all-sensibilities clue.

So please enjoy some immensely clever clues that didn’t make the cut elsewhere.

New York Times Crossword contributor Ian Livengood considers this one of his favorite answer/clue combinations, and it’s a real treat. He offers “Group getting some air play?” as the clue for “MILE HIGH CLUB.”

Patti Varol, Los Angeles Times Crossword contributor, turned some heads with the clue “Car bomb?” for “EDSEL.”

Crossword fiend Doug Peterson, via Patti, shared another eye-popping clue: “I had ‘BUILDING’ in a Sun puzzle and Peter Gordon clued it as ‘Erection’. We agreed that that one is hilarious and unlikely to be used anywhere else.”

My chums from Penny Press came through as well, offering clues that caught the attention of The Powers That Be.

Puzzle editor Keith Yarbrough contributed two wonderfully wicked clues for common crossword fodder, offering “Public hanging” for “ART” and “Brest milk” for “LAIT.”

Crossword guru Eileen Saunders also lobbed two sharp examples of choice wordplay my way, offering “Hoe house?” for “SHED” and “Wombmates?” for “TWINS.”

But my favorite contribution was definitely the one offered by variety puzzler Leandro Galban: “Kindergarten snack?” for “PASTE.”

Not only did I laugh out loud at that clue, but there’s more. In the email accompanying the clue, Leandro said, “We received at least one letter questioning my sanity so you know it was a success.”

I’d like to thank my fellow puzzlers for some tremendous wordplay and a few hearty laughs today. Keep on pushing the envelope! You never know what clever clues they’ll conjure next.

Puzzles in Pop Culture: Futurama

Posted on by
2

Not so long ago, I wrote a post about cryptography in the real world, highlighting moments where codebreaking made a difference in crime solving and espionage, and sometimes changed the course of history.

And while the encryptions featured in today’s entry aren’t quite as world-changing, they just as interesting.

I’m talking about the alien languages that were featured in the background of the animated television show Futurama.

At least two ciphers have been employed by the writers and animators of the show — a third is rumored to have appeared in the fourth season of the show, but there hasn’t been confirmation of that — and they’ve proven to be an engaging Easter egg for puzzle fans.

The first is called Alien Language One, or Alienese, and it appeared in the background of the show from the pilot episode onward. It’s a simple one-to-one code, with symbols for all 26 letters and 10 digits in standard English. (Supposedly it was solved by some enterprising puzzlers within a half-hour of the show’s premiere.)

A second, far more complex encryption started appearing during the show’s second season, and it’s called Alien Language Two, or Alienese II, and it’s based on an autokey cipher.

Autokey ciphers are more involved than a standard encryption, because there’s no one-to-one organizational structure. Instead, the symbol for a given letter or number can change based on the symbol that precedes it.

I’ll let the folks at the Futurama Wiki explain:

Each symbol has a numerical value. To decode a message, the first symbol’s value is translated directly into a character (0=’A’, 1=’B’, and so on). For the remaining letters, you subtract the previous symbol’s numerical value. If the result is less than zero, you add 26. Then that number is converted into a character as before.

This is some high-level puzzling, considering it’s a background joke-delivery system on an animated show. (But, considering the show does jokes about Schrodinger and throwaway gags based on mathematical principles like taxicab numbers, I’m not at all surprised.)

Of course, those puzzle-lovers at The Simpsons couldn’t help but get in on the fun, using Alienese as a background gag in a reference to the show Lost.

The masterminds at Futurama are definitely puzzlers at heart, and more than worthy of recognition in the Puzzles in Pop Culture library.

PuzzleNation Book Review: Mr. Penumbra’s 24-Hour Bookstore

Posted on by
4

Welcome to the fourth installment of PuzzleNation Book Reviews!

All of the books discussed and/or reviewed in PNBR articles are either directly or indirectly related to the world of puzzling, and hopefully you’ll find something to tickle your literary fancy in this entry or the entries to come.

Let’s get started!

Our book review post this time around features Robin Sloan’s novel Mr. Penumbra’s 24-Hour Bookstore.

Clay Jannon just lost his web-development job, and on a whim, he stumbles into a strange bookstore looking for a new night clerk. With stories-high shelves loaded with strange books that aren’t for sale, Mr. Penumbra’s 24-Hour Bookstore is more like a library for an eccentric smattering of visitors who come and go at all hours, checking out a single book at a time for reasons that elude Clay.

As Clay’s small circle of friends is drawn into the mystery of Mr. Penumbra’s store, Clay discovers a curious pattern dictating which book each visitor will select next, unintentionally taking the first step into tackling far greater and more peculiar secrets.

Whimsical yet still grounded in a believable world, Mr. Penumbra’s 24-Hour Bookstore is never what you expect. Wrapped in layers upon layers of curiosities — a puzzle within a puzzle about a puzzle, you could say — the book incorporates elements of urban fantasy, coming-of-age stories, and old-school mysteries to tell a wonderful story about the irresistable allure of a quest.

Every character in the book, from Clay’s friends to Mr. Penumbra’s patrons, falls into a situation plenty of puzzle devotees are quite familiar with: confronting a puzzle that seems unsolvable. That thread, that idea of a challenge awaiting if you’re just clever enough to conquer it, is a tantalizing one, and it drives a good chunk of the book’s plot.

What’s truly engaging about Sloan’s story, though, is its willingness to acknowledge the potential for disappointment from both sides. After all, some puzzles are too tough, some answers will elude us, and that’s part of what makes the challenge so enticing. But there’s also the disappointment that can follow victory. After all, if the unsolvable puzzle turns out to be solvable after all, will you be satisfied with the answer? And what comes next?

This double-bladed sword of possibilities elevates an already-intriguing plot and a thoroughly likable cast of characters into something truly enjoyable. This is a book rich in detail, setting, and charm, and even in the slow moments, your interest never flags.

The modern setting also added depth to both the mystery and how the characters confronted it. The book could’ve easily condemned paper books as old fashioned or e-readers as an obnoxious affront, but instead, it charts the highs and lows of the crossroads between print publishing and electronic media, allowing the best of both worlds to shine as the characters delve deeper into the mysteries surrounding Mr. Penumbra’s bizarre bookshop.

You know, the puzzle world can seem like such a secret society sometimes, working by its own weird rules and internal logic, and to see a book tackle that idea with such charm and lightness was a real treat. I look forward to seeing what Robin Sloan cooks up next.

Well, that’s it for the latest installment of PuzzleNation Book Reviews. I hope you enjoyed the post and look forward to more book discussions in the future. In the meantime, keep calm, puzzle on, and I’ll catch you later.