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.

longdiv1

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.

longdiv2

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.

longdiv3

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.

longdiv4

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:

longdiv5

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.

longdiv6

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.

longdiv7

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.

longdiv8

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

longdiv9

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.]


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!

Puzzles in Pop Culture: Gotham

[Image courtesy of Comic Book Movie.]

Fox’s comic book crime drama Gotham returns from its winter break on Monday, January 16th, kickstarting the second half of its third season. For the uninitiated, Gotham is set before the days of Batman’s adventures; Bruce Wayne is a young man, James Gordon is a detective, the city is rife with corruption, and most of Batman’s rogue’s gallery of enemies have yet to appear in the city.

Given that the show features one of the most infamous puzzly villains in history — The Riddler, aka Edward Nygma — I thought I would look back on the episode where the show’s version of the villain truly took shape.

So, in today’s blog post, we look back on season two’s “Mad Grey Dawn,” the episode that took the character beyond occasional riddles and into true Batman-style puzzly evil. (It’s worth noting that Edward Nygma’s day job is forensic scientist working for the Gotham City Police Department.)

We open in an art gallery, where a sculpture shaped like a bomb — that, curiously enough, is labeled “this is a real bomb” — rests in the center of the room. Nygma lights the fuse, chasing everyone out, then steals a painting, leaving behind a green question mark spray-painted in its place.

Detective Jim Gordon and his partner, detective Harvey Bullock, are assigned to the case. Before they go investigate, Gordon is held back by his captain, Nathaniel Barnes, who tells him an anonymous tip implicates him in a closed murder case, and Internal Affairs will be investigating.

At the art museum, they learn about the stolen painting, Mad Grey Dawn, which details a railway explosion. They discover two other, more valuable paintings were vandalized with spray-painted question marks, one by Gerard Marché and another by Henri Larue. Gordon believes the thief is trying to send a message, not trying to strike it rich.

And we have our first question: what’s the message?

[Image courtesy of EW.]

The viewer isn’t given much time to ponder it before Gordon realizes the artists ARE the clue. Marché is French for “market” and Larue is French for “the road.” They deduce that the thief is targeting the railway station on Market Street.

We then see Nygma removing a bomb in a bag from his car.

At the train station, Nygma is waiting. But Bullock and Gordon arrive as an order goes out to evacuate the building. Gordon spots a question mark spray-painted on a locker, and as soon as he does, Nygma remotely activates the timer on the bomb.

Gordon uses a crowbar to pry open the locker and get ahold of the bomb. Bullock and the other officers clear out the station and Gordon tosses the bomb before it explodes.

[Image courtesy of TV Line.]

As they investigate the bombing, they find no clues or riddles waiting for them. But Nygma is there, and he has an officer named Pinkney sign an evidence form for him. He then talks to Gordon, feeling him out on what Gordon knows about the bomber, and Gordon makes him the lead on forensics for the case.

Gordon is at a loss as to who the thief/bomber is or what he wants. But the viewer is presented with a different puzzle. We know who the bomber is, and we know he wants to destroy Jim Gordon. But how? How do these pieces we’ve seen fit together?

If you’re an attentive viewer, you’ve already spotted two big clues to Nygma’s trap.

Later, Nygma visits Officer Pinkney at home. He then asks him what you call a tavern of blackbirds, before hitting him with a crowbar.

Gordon looks over evidence photos when Bullock calls with info. They find a payphone the bomber used to trigger the bomb. Gordon heads off to check it out, discovering Pinkney’s murdered body in the apartment next door.

As he checks on the fallen officer, Captain Barnes walks in. Barnes reveals that Pinkney sent him a message, wanting to talk about Gordon. Gordon tries to explain that he was following up on a lead in the bomber case and stumbled upon Pinkney’s body, but Barnes takes him into custody.

[Image courtesy of Villains Wiki.]

Down at the station, Gordon talks to Barnes. Barnes reveals a crowbar was found with Gordon’s fingerprints. Gordon realizes it’s the crowbar from the train station. (Clue #1 from earlier.) He mentions the forensics report and Bullock’s call, but Barnes found nothing about that in the report when he checked it.

Barnes then reveals that Pinkney was the anonymous tip that reopened the murder case mentioned earlier, and he has a signed form to prove it. Amidst all these accusations, we see flashbacks of Nygma taking the crowbar, Nygma securing Pinkney’s signature at the bank (Clue #2 from earlier), and Nygma swapping out forensic reports in Bullock’s file.

Gordon has been thoroughly trapped in The Riddler’s web, and Barnes takes him in. Gordon is charged with the murder and sent to prison.

[Image courtesy of Comic Vine.]

Now, there aren’t the usual riddles to solve like you might expect (though there are plenty of riddles in earlier episodes). For puzzle fans, this episode is more about trying to unravel Nygma’s plan to stop Gordon while it unfolds. Did you manage it?

And the clues are all there, with the camera lingering on the crowbar in the bucket at the train station, and the scene of Pinkney signing the form for Nygma. It’s both a well-orchestrated frame-up and a well-constructed how-dun-it for the viewer.

And with an episode looming entitled “How the Riddler Got His Name,” I expect we’ll see more strong moments from this puzzly villain in the future.


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!

It’s Follow-Up Friday: Parking Lot edition!

Welcome to Follow-Up Friday!

For those new to PuzzleNation Blog, Follow-Up Friday is a chance for us to revisit the subjects of previous posts and update the PuzzleNation audience on how these projects are doing and what these people have been up to in the meantime.

And today, in the spirit of yesterday’s post about The Riddler, I’d like to post a brain teaser for you to solve!

Can you explain the numbering system in this parking lot?

Also, do you have any favorite riddles or brain teasers? Let us know and we might feature them in a future post!

Thanks for visiting PuzzleNation Blog today! You can share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and be sure to check out the growing library of PuzzleNation apps and games!

Riddle Me This, Batman!

With so much Dark Knight-related news floating around, this might very well be the Year of Batman.

Not only is it the 75th anniversary of Batman’s first appearance in comics, but he’s going to be all over screens in the coming months. There are already teaser images out for the 2016 Batman Vs. Superman film, plus Fox’s upcoming prequel drama Gotham. As if that wasn’t enough, it was recently announced that the entire 1960s television series is coming to DVD and Blu-ray!

So it’s the perfect opportunity to take a look at Batman’s puzzliest foe, The Riddler.

Played by Frank Gorshin (and John Astin, in one episode) in the 1960s TV series, The Riddler was absolutely manic, often breaking into wild fits of anger and laughter, compulsive in his need to send taunting riddles to the Caped Crusader. (This actually became a point of contention with his fellow villains in the Batman film, since they didn’t appreciate Batman being tipped off to their plans by the Riddler’s riddles.)

His riddles were similarly inconsistent and unpredictable. Some of them were genuine puzzlers (highlight for answers):

  • How do you divide seventeen apples among sixteen people? Make applesauce.
  • There are three men in a boat with four cigarettes but no matches. How do they manage to smoke? They throw one cigarette overboard and make the boat a cigarette lighter.

Others were similar to children’s jokes, silly in the extreme:

  • What has yellow skin and writes? A ballpoint banana.
  • What suit of cards lays eggs? One that’s chicken-hearted.

While Gorshin’s Riddler is probably the most famous and familiar version of the villain, my personal favorite was the Riddler in Batman: The Animated Series.

This Riddler (voiced by actor John Glover) was more suave, sophisticated, and cunning. In his debut episode, entitled “If You’re So Smart, Why Aren’t You Rich?”, he challenged Batman and Robin with a boobytrap-filled labyrinth, complete with numerous riddles that were far more challenging than Gorshin’s Riddler would ever use.

Fittingly, a final riddle — the Riddle of the Minotaur — awaits Batman at the center of the labyrinth. Can you solve it?

I have millions of eyes, but live in darkness.
I have millions of ears, but only four lobes.
I have no muscles, but I rule two hemispheres.
What am I? The human brain.

This version of the Riddler only appeared a few times in the animated series, due to the writers’ difficulty coming up with his riddles and keeping his complex plots simple enough to fit into a single episode’s runtime. Nonetheless, this Riddler remains a fan favorite.

In the film Batman Forever, Jim Carrey actually gives us three versions of the Riddler: the scientist Edward Nygma (who definitely has a screw or two loose), the businessman Edward Nygma (a charming, public facade betraying none of his truly villainous tendencies), and the Riddler (a diabolical, borderline insane villain with an ax to grind with Bruce Wayne). Whether you like Carrey’s take on the character or not, he embodies an effective combination of Gorshin and Glover’s Riddlers.

Of course, despite his bizarre plan to read people’s minds through television and become smarter by doing so, this Riddler had one advantage over the previous versions: he had puzzlemaster Will Shortz designing his riddles for him.

While each riddle by itself meant little, except for providing a challenge to Batman’s intellect, all of them combined revealed the Riddler’s identity and set up the final showdown between the heroes and villains.

As for the newest version of the Riddler in Fox’s fall premiere Gotham, we don’t know a lot about him, since this is set in the days before Batman prowled the streets.

Apparently, the younger Edward Nygma (played by theater actor Cory Michael Smith) is a forensic scientist working for the police (who may formulate his notes in the form of riddles).

Hopefully, this Riddler will carry on the fine tradition of Riddlers past, challenging the defenders of Gotham to outthink him, rather than outfight him.


Thanks for visiting PuzzleNation Blog today! You can share your pictures with us on Instagram, friend us on Facebook, check us out on TwitterPinterest, and Tumblr, and be sure to check out the growing library of PuzzleNation apps and games!