Brain Teaser: A Curious Way to Tell Time

We love tackling brain teasers, riddles, math puzzles, and mind ticklers here at PuzzleNation Blog.

Our friends at Penny Dell Puzzles also enjoy putting their puzzly skills to work by posting various brain teasers..

And one of our mutual readers was hoping we could explain how to solve a puzzle that recently made the rounds on Penny’s social media accounts.

We’d be happy to!

Today, let’s take a look at a brain teaser all about time.

So, where do we begin?

Let’s start by breaking that sentence down to make it easier to parse: “The number of hours left today is half of the number of hours already passed.”

Well, a simpler way of saying that is “the number of hours already passed is twice the number of hours left.”

So if you have the number of hours left — let’s call that X — then the number of hours already passed is twice X, or 2X. Between X and 2X, that’s your entire day covered.

Sorting that out gives us the simple formula of X + 2X = 24, since there are 24 hours in a day.

That easily becomes 3X = 24, and simple division tells us that X = 8.

So X, the number of hours left today, is 8. Which means that twice X, or 16, is the number of hours already passed.

And if there are 8 hours left in the day (or 16 hours already passed), that means it’s 4 PM.

Most of the time, brain teasers are all about efficiently organizing the information we have.

That allows us to figure out how best to use that information to move forward and solve the puzzle. This is just as true with a relatively straightforward brain teaser like this as it is with a complicated logic puzzle with all sorts of pieces to put together.

It’s all about figuring out what we know, what we don’t, and how what we DO know can lead us to what we don’t.

That’s just part of the fun.

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Answers to Our 6th Anniversary Instagram Brain Teasers!

Last week, we celebrated six years of PuzzleNation Blog by announcing a week-long puzzly social media blitz.

Facebook and Twitter saw twice-daily alerts for the puzzle of the day for both Daily POP Crosswords and Penny Dell Crosswords App, cuing solvers to contact us with the answers to particular across and down clues.

Instagram solvers were encouraged to tackle a series of brain teasers, and today, we’ve got all the answers for you! Let’s jump right in.

We started off on Tuesday with this relatively straightforward brain teaser: How can you add eight 4s together so that the total adds up to 500?

We got the most responses to this one, and it’s no surprise, as we have some very crafty followers on Instagram. The trick here is number placement. By grouping 4s, you create larger numbers that make it easier to add to your total.

Solution: 444 + 44 + 4 + 4 + 4 = 500

Wednesday’s puzzle involved placing the numbers 1 through 8 into the grid above. Consecutive numbers cannot appear in an adjacent or diagonal box.

This puzzle was actually created and submitted by a PuzzleNationer named Sanjana, so kudos to you, Sanjana, as you made one heck of a brain teaser!

Here’s the solution. (Using the same numbers in reverse or flipped layout creates four different variations on the same solution.)

Thursday’s brain teaser put your Scrabble and Upwords skills to the test, as we played a round of Quad-Doku! The goal is to play each tile, one at a time, onto the board, forming a new common word (or words) each time. Do this with all 8 tiles in any order. By the end, all four corners will have changed.

This is a nice chain-solving puzzle, and here’s the solution we came up with:

F makes FOUR/FIND, S makes FINS/SEEM, A makes SEAM, B makes FIBS, C makes SCAM, W makes SWAM, L makes FOUL/LOOM, and P makes LOOP/SWAP.

On Friday, we posted a riddle to test your puzzly skills. Once I am 24, twice I am 20, three times I am unclean. What am I?

Solution: The answer is X. It’s the 24th letter of the alphabet, two X’s makes 20 in Roman numerals, and three X’s marks something as inappropriate for some viewers.

Monday brought us our final brain teaser, a matchstick puzzle (or, in this case, a toothpick puzzle). Can you move four toothpicks in order to change the zigzag path into 2 squares? The two squares do not have to be equal in size.

In the image above, we’ve circled the four toothpicks to move.

And here is the completed puzzle, with two squares of unequal size.

How did you do, intrepid solvers? Well, based on the responses we received, pretty darn well! We’ll be reaching out to contest winners later this week!

But in the meantime, we’d like to thank everyone who participated in our PN Blog 6th Anniversary event. You help make this the best puzzle community on the planet, and we are forever grateful.

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PuzzleNation Blog Looks Back on 2017!

2017 is quickly coming to a close, and as I look back on an eventful year in the world of puzzles and games, I’m incredibly 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 game designers, constructors, artists, and creative types of all kinds.

We unraveled math puzzles and tackled the Crossword from Hell. We accepted the challenge of diabolical brain teasers, optical illusions, Internet memes, and more, even pondering our place in the world of puzzles as electronic solvers like Dr. Fill and AlphaGo rise in capability.

We delved into puzzle history with posts about the legacy of female codebreakers in World War II, game dice from centuries ago, theories about Shakespeare’s secret codes, and the long history of cryptography and the NSA. We brought to light valuable examples of puzzles in art, popular culture, famous quotations, and even the natural world as we pondered whether bees are verifiable problem-solvers like crows and octopuses.

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 worthy causes like Puzzles for Progress, as well as amazing projects like new escape rooms, dazzling corn mazes, and the ongoing Kubrick’s Game interactive experience.

We cheered the 75th anniversary of the New York Times Crossword, and chronicled the many celebrations that marked the occasion, from guest crossword constructors like Bill Clinton and Lisa Loeb to a puzzle-centric cruise across the Atlantic!

We also mourned as friends and fellow puzzlers passed on. We said goodbye to David Lindsey and Raymond Smullyan, two underappreciated giants of the field. The pun-fueled show @midnight this year, which inspired our monthly hashtag game, also closed up shop, sadly.

We celebrated International TableTop Day, made puzzly bouquets in honor of International Puzzle Day, marveled at the records broken at the Rubik’s Cube World Championship, attended the American Crossword Puzzle Tournament and New York Toy Fair, and dove deep into an ever-expanding litany of puzzle events like the Indie 500, BosWords, Lollapuzzoola 10, and Crosswords LA.

We found puzzly ways to celebrate everything from Pi Day, the Super Bowl, and Star Wars Day to Halloween, Thanksgiving, and Christmas, and we were happy to share so many remarkable puzzly landmark moments with you. We even discovered Puzzle Mountain!

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 five years of PuzzleNation Blog this year, I recently penned my 800th 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 2017 went well beyond that.

Every month, we delivered quality content for the Penny Dell Crosswords App. From monthly deluxe sets and bonus boxes to Dell Collection sets and holiday bundles, dozens upon dozens of topnotch puzzles wended their way to our loyal and enthusiastic solvers.

And just last month, we launched our newest puzzly endeavor — Daily POP Crosswords — bringing you fresh, up-to-date cluing and relatable themes in world-class puzzles created by some of the industry’s best constructors! (Many of whom you’ve gotten to know in our recent interview series, Meet the Daily POP Crosswords Constructors!)

But whether we’re talking about the Penny Dell Crosswords App or Daily POP Crosswords, I’m proud to say that every single puzzle represents our high standards of quality puzzle content crafted for solvers and PuzzleNationers.

And your response has been fantastic! Daily POP Crosswords is thriving, the blog has over 2200 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.

2017 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. The new year looms large, and we look forward to seeing you in 2018!

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The Diabolical Long Division Brain Teaser!

From time to time, I’ll receive an email with a brain teaser I’ve never seen before. Sometimes they come from friends, or fellow puzzlers. Other times, PuzzleNationers will ask for my assistance in solving a puzzle that has flummoxed them.

That was the case with today’s puzzle, and I’ll admit, this one was a bit of a doozy to unravel.


Yup, an entire long division problem with only a single digit set. No letters or encryption to let us know which digits were repeated, as there are in Word Math puzzles published by our friends at Penny Dell Puzzles.

Just a 7 and a bunch of asterisks. “Is this doable?” the sender asked.

Yes, this is entirely doable, friend. Let’s break it down step by step.

First, we need to know our terminology. The 8-digit number being divided is our dividend. The 3-digit number we’re dividing into it is the divisor. The 5-digit number on top is our quotient.

For the other lines, let’s label them A through G for ease of reference later.


There we go. Now, where do we go from here? We start with what we know.

We know that 7 is the second digit in our quotient.

So our divisor, times 7, equals the number on line C. That’s a 3-digit number, which means the first number in our divisor is 1. Why? Because if it was 2, 2 times 7 would give us 14, which would be a 4-digit number on that line.


That means the quotient is somewhere between 100 and 142. (Why 142? Easy. I divided 1000 by 7, and 142 is the last 3-digit number you can multiply 7 against and still end up with a 3-digit answer for line C. 143 times 7 is 1001, which is too high.)

What else do we know from the puzzle as it stands?

Well, look at lines E and F. We bring both of the last two digits in the dividend down for the final part of the equation. What does that mean?

Remember how long division works. You multiply the divisor by whatever number gets you closest to the given digits of the dividend, subtract the remainder, bring down the next digit from the dividend, and do it all over again until you get your answer.

You multiply the first digit of the quotient times the divisor to get the number on line A. You multiply 7 times the divisor to get the number on line C. You multiply the third digit of the quotient times the divisor to get the number on line E.

Following this route, you would multiply the fourth digit of the quotient against the divisor to get the number on line G. But bringing just one digit down didn’t give us a number high enough to be divided into. Instead of needing more lines (H and I, in this case), we bring the last digit of the dividend down and press onward.

That means the fourth digit of the quotient is 0, because the divisor went into the dividend zero times at that point.


And there’s more we can glean just from the asterisks and what we already know. We know that every one of those 4-digit numbers in the equation begin with the number 1.

How do we know that? Easy. That first number in the divisor. With a 1 there, even if the divisor is 199 and we multiply it times 9, the highest possible answer for any of those 4-digit numbers is 1791.

So let’s fill those numbers in as well:


Now look at lines D, E, and F. There’s nothing below the 1 on line D. The only way that can happen is if the second digit in line D is smaller than the first digit on line E. And on line F, you can see that those first two columns in lines D and E equal zero, since there’s nothing on line F until we hit that third column of digits.

That means the second digit on line D is either a 0 or a 1, and the first digit on line E is a 9. It’s the only way to end up with a blank space there on line F.


I realize there are a lot of asterisks left, but we’re actually very close to knowing our entire quotient by now.

Look at what we know. 7 times the divisor gives us a 3-digit answer on line C. We don’t yet know if that’s the same 3-digit answer on line E, but since it’s being divided into a 4-digit number on line E and only a 3-digit number on line C, that means the third digit in our quotient is either equal to or greater than 7. So, it’s 7 or 8.

Why not 9? Because of the 4-digit answers on lines A and G. Those would have to be higher than the multiplier for lines C and E because they result in 4-digit answers, not 3. So the digit in the first and fifth places in the quotient are higher than the digit in the third. So, if the third digit in the quotient is 7 or 8, the first and fifth are either 8 or 9.

So how do we know whether 7 or 8 is the third digit in the quotient?

Well, if it’s 7, then lines C and E would have the same 3-digit answer, both beginning with 9. But line C cannot have an answer beginning with 9, because line B is also 3 digits. The highest value the first digit in line B could have is 9, and 9 minus 9 is zero. But the number on line D begins with 1, ruling out the idea that the numbers on lines C and E are the same.

That makes the third digit in the quotient 8, and the first and fifth digits in the quotient 9.


We know our quotient now, 97809. What about our divisor?

Well, remember before when we narrowed it down to somewhere between 100 and 142? That’s going to come in handy now.

On line F, we know those first two digits are going to be 141 or below, because whatever our divisor is, it was larger than those three digits. That’s how we ended up with a 0 in our quotient.

So, the number on line D minus the number on line E equals 14 or below. So we need a 900-something number that, when added to a number that’s 14 or below, equals 1000 or more. That gives us a field from 986 to 999.

And that number between 986 and 999 has to be divisible by 8 for our quotient to work. And the only number in that field that fits the bill is 992. 992 divided by 8 gives us 124, which is our divisor.


From that point on, we can fill out the rest of the equation, including our lengthy dividend, 12128316.


And there you have it. With some math skills, some deduction, and some crafty puzzling, we’ve slain yet another brain teaser. Nice work everyone!

[After solving the puzzle, I did a little research, and apparently this one has been making the rounds after being featured in FiveThirtyEight’s recurring Riddler feature, so here’s a link.]

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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]

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.

Thanks for visiting PuzzleNation Blog today! Be sure to sign up for our newsletter to stay up-to-date on everything PuzzleNation!

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!