Fix These Sixes!

[Image courtesy of Daily Brain Teaser.]

It’s a new year, and I’ve already got a new math puzzle for you. A friend discovered this one and sent it my way in the hopes that I’d be able to crack this diabolical brain teaser.

Fellow puzzlers and PuzzleNationers, this is the Sixes puzzle.

0 0 0 = 6

1 1 1 = 6

2 2 2 = 6

3 3 3 = 6

4 4 4 = 6

5 5 5 = 6

6 6 6 = 6

7 7 7 = 6

8 8 8 = 6

9 9 9 = 6

The goal is to make all of the above equations true by adding mathematical symbols.

And, naturally, there are rules.

  • You can use as many mathematical symbols as you want for each equation
  • You are not allowed to use letters, so spelling out functions like “cos” is out
  • You are not allowed to add digits of any kind, so no turning a “2” into a “12”
  • The result has to be exactly 6 ( not 6.0000000000000001 or 5.999999999999999 )
  • Square roots are allowed
  • You are not allowed to change “=” to “≠” (not equal to) or manipulate the result in any way

Take a crack at it, then scroll down past the dice to see how it’s done.

Last warning before answers!

[Image courtesy of The Progzilla Files.]

Let’s start by knocking out the easiest one.

2 2 2 = 6 can be resolved as 2+2+2=6.

6 6 6 = 6 can be resolved as 6+6-6=6 or 6-6+6=6.

They get a little trickier from here, involving multiple operations.

3 3 3 = 6 can be resolved as 3×3-3=6, becoming 9-3=6.

Fractions also come into play for a few of these equations. (This can also be represented as division.)

5 5 5 = 6 can be resolved as 5+(5/5)=6, becoming 5+1=6.

7 7 7 = 6 can be resolved as 7-(7/7)=6, becoming 7-1=6.

[Image courtesy of TAF.org.]

Okay, we’re halfway there, and now the square root rule gives us a hint regarding how to resolve the 4 equation.

4 4 4 = 6 can be 4+4-(√4)=6, becoming 8-2=6.

And this formula gives us a way to crack the 8 equation.

8 8 8 = 6 can be resolved as 8-√(√(8+8))=6, becoming 8-√(√16)=6, becoming 8-(√4)=6, becoming 8-2=6.

Square roots also come into play in solving the 9 equation.

9 9 9 = 6 can be resolved as (9+9)/(√9)=6, becoming 18/3=6.

Now, admittedly, at this point, I was stumped. I had two equations left, and no ideas regarding how to proceed.

0 0 0 = 6

1 1 1 = 6

So, I reached out to a mathematician pal of mine — the same one who helped me crack the diabolical Seesaw Puzzle from Brooklyn Nine-Nine — and he immediately knew what to do: use an exclamation point.

In mathematics, an exclamation point represents a factorial, the product of every positive number between the given number and zero.

For instance, 6! represents 6x5x4x3x2x1, or 720.

1 1 1 = 6 can now be resolved as (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

But what about 0 0 0 = 6?

Factorials to the rescue again! You see, 0! equals 1. So we can use the 1 equation as a template for this one.

0 0 0 = 6 can be resolved as (0!+0!+0!)!=6, becoming (1+1+1)!=6, becoming 3!=6, becoming 3x2x1=6.

And there you have it, the Sixes puzzle conquered with nothing but crafty math and puzzly skills. An excellent start to a new year of brain teaser challenges!


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It’s Follow-Up Friday: 2016 Countdown edition!

It’s the final Follow-Up Friday of the year, so what do you say we revisit all of 2016 with a countdown of my ten favorite blog posts from the past year!


#10 Doomsday Prep

One of the big surprises for me this year was discovering that crosswords and puzzle books were hot-ticket items for doomsday preppers. The idea that crosswords belong next to necessities like food, water, shelter, and knowledge was a revealing one, something that gave me great hope for the future, whether we need those caches or not.

#9 Holiday Puzzly Gift Guide

Every year, one of my favorite activities is putting together our Holiday Puzzly Gift Guide. I get to include the best products sent to me for review by top puzzle and game companies, mix in some of my own favorites, and draw attention to terrific constructors, game designers, and friends of the blog, all in the hopes of introducing solvers (and families of solvers) to quality puzzles and games.

#8 A Puzzly Proposal

Our friends at Penny Dell Puzzles once again pulled off a heck of a puzzly coup when an intrepid fellow puzzler asked them for help proposing to his girlfriend with a special Simon Says puzzle.

I reached out to the lucky fiancé and got his permission to share the story with the PuzzleNation readership, and as I learned more about who was involved and how they’d managed to make it happen, I enjoyed the story more and more. Here’s hoping for many happy puzzly years ahead for the young couple!

#7 Puzzle Fort

For International Puzzle Day, I built a fort out of puzzle books.

It was awesome. Definitely one of my favorite puzzly moments of the year.

#6 The End of Sudoku?

The Sudoku boom may be over, but Sudoku remains one of the most popular puzzles in the world, and I got to thinking… when would we run out? I mean, eventually, statistically speaking, every single Sudoku puzzle permutation would get used at some point, so when would that happen?

So, I crunched the numbers, and it turns out, we’ve got centuries before that happens. Still, it was a fun mental puzzle to unravel.

#5 Murder Mystery

At some point this year, I let slip to my fellow puzzlers that I’d written and staged murder mystery dinners in the past, but it had been a while since I’d done anything like that. Naturally, they volunteered to be participants, urging me to stage something in the office.

Eventually, I accepted their challenge, pitting myself against a half-dozen or so of my fellow puzzlers, allowing some of them to investigate while others played a part in the mystery. It was an enormous undertaking and an absolute blast that lasted three days, and it was definitely a highlight of the year for me.

#4 Puzzle Plagiarism

There was probably no bigger story in crosswords all year than the accusations of plagiarism leveled against Timothy Parker. The editor of puzzles for USA Today and Universal UClick. After numerous examples of very suspicious repetitions between grids were discovered in a crossword database compiled by programmer Saul Pwanson and constructor Ben Tausig, Parker “temporarily stepped back from any editorial role” with their puzzles.

Eventually, Parker was removed from any editorial influence on USA Today’s puzzles, but it remains unknown if he’s still serving in a puzzle-related capacity for Universal Uclick. But the real story here was about integrity in puzzles, as many puzzle and game companies rallied to defend their rights as creators. That’s a cause we can all get behind.

#3 Interviewing the PuzzleNation Team

Our recurring interview feature 5 Questions returned this year, but what made it truly special to me was being able to turn the spotlight on some of my fellow puzzlers here at PuzzleNation as part of celebrating 4 years of PuzzleNation Blog. Introducing readers to our programmer Mike, our Director of Digital Games Fred, and yes, even myself, was a really fun way to celebrate this milestone.

#2 ACPT, CT FIG, and Other Puzzly Events

There are few things better than spending time with fellow puzzlers and gamers, and we got to do a lot of that this year. Whether it was supporting local creators at the Connecticut Festival of Indie Games or cheering on my fellow puzzlers at the American Crossword Puzzle Tournament, getting out and talking shop with other creators is invigorating and encouraging. It really helps solidify the spirit of community that comes with being puzzly.

#1 Penny Dell Sudoku and Android Expansion

Those were our two biggest app releases this year, and I just couldn’t choose one over the other. This has been a terrific year for us as puzzle creators, because not only did we beef up our library of Android-available puzzle sets to match our terrific iOS library, but we launched our new Penny Dell Sudoku app across both platforms, broadening the scope of what sort of puzzle apps you can expect from PuzzleNation.

It may sound self-serving or schlocky to talk about our flagship products as #1 in the countdown, but it’s something that we’re all extremely proud of, something that we’re constantly working to improve, because we want to make our apps the absolute best they can be for the PuzzleNation audience. That’s what you deserve.

Thanks for spending 2016 with us, through puzzle scandals and proposals, through forts and festivities, through doomsday prepping and daily delights. We’ll see you in 2017.


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PuzzleNation Looks Back at 2016!

The year is quickly coming to a close, and as I look back on an eventful year in the world of puzzles and games, I’m unbelievably proud of the contributions both PuzzleNation Blog and PuzzleNation made to the puzzle community as a whole.

Over the last year, we explored board games and card games, strategy games and trivia games, dice games and tile games, do-it-yourself puzzlers and pen-and-paper classics. We met designers, constructors, authors, artists who work in LEGOs and dominos, and creative types of all kinds.

We unraveled math puzzles and used statistics to play Hangman and Guess Who smarter. We accepted the challenge of diabolical puzzles, optical illusions, Internet memes, and more.

We delved into puzzle history with posts about Bletchley Park, puzzle graffiti from ancient Greece, Viking board games, and modern mysteries like the Kryptos Sculpture and the Voynich Manuscript. We separated fact from fiction when it comes to puzzles and brain health, avoiding highfalutin promises and sticking to solid science.

We spread the word about numerous worthwhile Kickstarters and Indiegogo campaigns, watching as the puzzle/game renaissance continued to amaze and surprise us with innovative new ways to play and solve. We shared amazing projects and worthy causes like Humble Bundles and puzzle/game donation programs for schools that allowed puzzle lovers to help others.

We celebrated International TableTop Day, built a puzzle fort in honor of International Puzzle Day, attended the American Crossword Puzzle Tournament and the Connecticut Festival of Indie Games, and dove deep into puzzle events like the Indie 500, the UK Sudoku Championship, the 2016 UK Puzzle Championship, and Lollapuzzoola. We even celebrated a puzzly wedding proposal, and we were happy to share so many remarkable puzzly landmark moments with you.

It’s been both a pleasure and a privilege to explore the world of puzzles and games with you, my fellow puzzle lovers and PuzzleNationers. We marked four years of PuzzleNation Blog this year, I’m approaching my 650th blog post, and I’m more excited to write for you now than I was when I started.

And honestly, that’s just the blog. PuzzleNation’s good fortune, hard work, and accomplishments in 2016 went well beyond that.

In April, we launched Penny Dell Crosswords Jumbo 3 for iOS users, and in May, we followed that with Penny Dell Crosswords Jumbo for Android. In November, we launched our new Penny Dell Sudoku app on both Android and iOS.

But the standout showpiece of our puzzle app library remains the Penny Dell Crossword App. Every month, we release puzzle sets like our Dell Collection sets or the themed Deluxe sets for both Android and iOS users, and I’m proud to say that every single puzzle represents our high standards of quality puzzle content for solvers and PuzzleNationers.

We even revamped our ongoing Crossword Clue Challenge to feature a clue from each day’s Free Daily Puzzle in the Crossword app, all to ensure that more puzzle lovers than ever have access to the best mobile crossword app on the market today.

And your response has been fantastic! The blog is closing in on 2000 followers, and with our audience on Facebook, Twitter, Instagram, and other platforms continuing to grow, the enthusiasm of the PuzzleNation readership is both humbling and very encouraging.

2016 was our most ambitious, most exciting, and most creatively fulfilling year to date, and the coming year promises to be even brighter.

Thank you for your support, your interest, and your feedback, PuzzleNationers. Have a marvelous New Year. We’ll see you in 2017!


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The GCHQ Strikes Again!

Last year, one of the toughest puzzles I encountered all year was the GCHQ Christmas Card.

The GCHQ — Government Communications Headquarters — provides security and intelligence services for the British government. Back when they were known as GC&CS — Government Code and Cypher School — they were responsible for funding Bletchley Park and its successes cracking the German “Enigma” code during World War II.

And last Christmas, they released a puzzly Christmas card that challenged the staunchest puzzlers, with over 600,000 submissions, but only 3 successful solutions!

This year, they’re doing things a little bit differently.

[Click here for a larger version.]

This puzzle is the first step in a larger event that the GCHQ expects will take MONTHS to solve. (Their official due date for submissions is February 28, 2017!)

The big change here is that instead of a series of webpages available free to the public, they’re releasing this puzzly master challenge as a book containing over 140 codes and puzzles.

But it’s for a good cause, as “all GCHQ’s proceeds from sales of the book will be donated to the Heads Together campaign, supported by HRH The Duchess of Cambridge, who wrote the foreword for The GCHQ Puzzle Book.”

The Duchess writes:

I have always been immensely proud of my grandmother, Valerie Glassborow, who worked at Bletchley Park during the Second World War. She and her twin sister, Mary, served with thousands of other young women as part of the great Allied effort to break enemy codes. They hardly ever talked about their wartime service, but we now know just how important the men and women of Bletchley Park were, as they tackled some of the hardest problems facing the country.

In a new century, their successors at GCHQ continue this intellectual tradition. Like their Bletchley predecessors, they have become well known for valuing and understanding the importance of mental wellbeing. This is so important when dealing with such discretion and the pressure which comes with this.

William, Harry and I are very grateful that this book is supporting our Heads Together Campaign. I hope it will not only amuse and challenge readers, but help to promote an open discussion of mental health problems, which can affect anyone, regardless of age or background. Together, we are aiming to change the national conversation around mental health from stigma and fear to openness and understanding. Those who buy this book and support the Heads Together campaign will be playing a part in helping people get the important mental health care they deserve.

Puzzles and charitable works: a perfect holiday match, to be sure.

For US shoppers, the book is available as a Kindle ebook through Amazon.com, but if you want a paperback copy, you’re better off ordering it from Amazon.co.uk. With shipping and conversion, it still cost me less than $20.


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It’s Follow-Up Friday: Blind Spot edition!

[No, not THAT Blindspot. Though friend of the blog David Kwong works on that show…]

Welcome to Follow-Up Friday!

Follow-Up Friday is a chance for us to revisit the subjects of previous posts and bring the PuzzleNation audience up to speed on all things puzzly.

And today I’d like to return to the subject of optical illusions.

[Image courtesy of Geekologie.]

We’ve discussed optical illusions many times in the past, covering everything from the Necker Cube to the Dress, but today’s illusion is a little bit different.

The 12-dot optical illusion pictured above, also known as a Hermann grid, has been making the rounds lately, and although many puzzlers have accused people of using a video or an animated gif instead of a simple picture, that’s simply not true.

This is a static picture, but our eyes are not designed to capture 12 dots in view at once, so the others vanish when you focus on one dot in particular.

And the folks at ASAP Science have an explanation for this phenomenon! As it turns out, this illusion and others involve both how your brain processes visual information and how the natural structure of your eye creates a blind spot that some optical illusions exploit.

Check out the full explanation here:

So next time someone shares one of these optical illusions, you’ll be ready to explain how they work and show off your puzzly knowledge.


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You can also share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and explore the always-expanding library of PuzzleNation apps and games on our website!

Cracking Einstein’s Riddle!

Fellow puzzlers and PuzzleNationers, it’s time to put on our thinking caps and crack another puzzly mystery!

I’ve seen this riddle making the rounds online lately, and like many logic problems, puzzles, and brain teasers that go viral, it claims that 98% of the world couldn’t solve it!

Well, I don’t buy that. (I also don’t buy that Einstein created it or had anything to do with it, but since it circulates under the name “Einstein’s Riddle,” I’m also calling it such.)

[Image courtesy of Mental Floss.]

Einstein’s Riddle

  • There are five houses that are each a different color.
  • There is a person of a different nationality in each house.
  • The five owners drink a certain drink. They each smoke a certain brand of cigar and also have a certain pet. No owner has the same pet, smokes the same brand of cigars, nor drinks the same drink.

CLUES

1. The British man lives in the red house.
2. The Swedish man has a dog for a pet.
3. The Danish man drinks tea.
4. The green house is on the left of the white house.
5. The owner of the green house drinks coffee.
6. The person that smokes Pall Mall raises birds.
7. The owner of the yellow house smokes Dunhill.
8. The person that lives in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The person that smokes Blend lives next to the one that has a cat.
11. The person that has a horse lives next to the one that smokes Dunhill.
12. The one that smokes Bluemaster drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to a blue house.
15. The man that smokes Blend has a neighbor that drinks water.

The question is “who owns the fish?”


Now, the first step is going through the clues and getting all five options for every variable. This will help us with the second step: building a grid to help us organize information.

  • Colors: Blue, Green, Red, White, Yellow
  • Nationalities: British, Danish, German, Norwegian, Swedish
  • Beverages: Beer, Coffee, Milk, Tea, Water
  • Cigars: Blend, Bluemaster, Dunhill, Pall Mall, Prince
  • Pets: Birds, Cat, Dog, Fish Horse

Okay, let’s build our grid. Now, we could list every intersection of information, like a full logic problem grid, but I don’t think that’s necessary here. We can simplify.

So where do we start? Well, since several clues refer to a first house, a center house, or neighboring houses, let’s assume that we’re talking about five houses in a row, and use that as our top line. Then we can list all of the other categories we need to determine along the left side.

Now let’s fill in what we know from the clues. We know the Norwegian lives in the first house (clue #9), the man living in the center house drinks milk (clue #8), and the Norwegian lives next to the blue house (clue #14).

This might not seem like much, but we can already determine what color the Norwegian’s house is. The British man lives in the red house (clue #1), so the Norwegian’s house isn’t red. The green house is on the left of the white house (clue #4), so the Norwegian’s house is neither green nor white, since there’s no house to the left of the Norwegian’s house, and the house to the right is blue. Therefore, the Norwegian’s house is yellow, the only color left.

And that tells us something else. The owner of the yellow house smokes Dunhill (clue #7), so we know what the Norwegian smokes. Plus, the man who keeps the horse lives next to the man who smokes Dunhill (clue #11), so we know the pet for the blue house.

We can now determine the color for every house. The green house is to the left of the white house (clue #4), and the owner of the green house drinks coffee (clue #5). But since we already know the owner of the center house drinks milk, that means the green house has to be the fourth, and the white house the fifth. Therefore, the center house is red.

But that’s not all we know now! The British man lives in the red house (clue #1), so we can place him as well.

Hmmm, where do we go from here? Well, let’s take a look at the beverages. We know the Norwegian doesn’t drink coffee or milk, but we also know that the Danish man drinks tea (clue #3), so the Norwegian doesn’t. We also know that the one that smokes Bluemaster drinks beer (clue #12), and the Norwegian smokes Dunhill, so that eliminates beer. Therefore, the Norwegian drinks water.

Since the man that smokes Blend has a neighbor that drinks water (clue #15), we can place Blend in the second house. (Clue #10 tells us that the person that smokes Blend lives next to the one that has a cat, but right now, we can’t be sure if that’s the first house or the center house, so let’s table that clue for now.)

But remember clue #12, the one that smokes Bluemaster drinks beer? Well, we know all the drinks except the second house and the fifth house, and the owner of second house smokes Blend, so the owner of the fifth house must both smoke Bluemaster and drink beer.

That leaves tea as the only possible beverage for the second house, and the Danish man drinks tea (clue #3), so we can place him as well.

Now we’re cooking! The German smokes Prince (clue #13), and since we know the nationality of the third house’s owner and what the fifth house’s owner smokes, the only option remaining is the fourth house. By process of elimination, that also places the Swedish man in the fifth house and Pall Mall in the center house.

And we’re left with only the pets to place. Appropriate, given that the question that got us started is “who owns the fish?”

According to clue #2, the Swedish man has a dog for a pet, and the person that smokes Pall Mall raises birds, so that takes care of the center and fifth houses, leaving only the first house and the fourth.

Clue #10, the one we put aside earlier, now comes into play. The person that smokes Blend lives next to the one that has a cat, and since the neighbor on one side (the center house) raises birds, that leaves only the first house open to have a cat.

Therefore, the German man — who smokes Prince, drinks coffee, and lives in the green house — also owns a fish.

98% of people can’t solve it? Apparently, I have greater faith in solvers than “Einstein” did.


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You can also share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and explore the always-expanding library of PuzzleNation apps and games on our website!