A Barrage of Brain Teasers!

[Image courtesy of SharpBrains.com.]

One of our favorite pastimes here on PuzzleNation Blog is cracking brain teasers. From riddles to logic problems, we accept challenges from all comers, be they TV detectives or fellow PuzzleNationers!

I got an email a few days ago from a reader who needed help unraveling a few brain teasers from a list she found online. She was proud to have solved most of them, but a few had eluded her.

We’re always happy to assist a fellow puzzler, so let’s take a look at those brain teasers!


In case you want to try them for yourself before we reveal the answers and how to solve each puzzle, I’ll list the original puzzles here and put a nice spoiler-safe break between the questions and the answers.

QUESTION 1: If you have a 7-minute hourglass and an 11-minute hourlass, how can you boil an egg for exactly 15 minutes?

QUESTION 2: Name the next number in the following sequence: 1, 11, 21, 1211, 111221, 312211, _____.

QUESTION 3: Four people want to cross a river, but the only option is a narrow bridge. The bridge can only support two people at a time. It’s nighttime, and the group has one torch, which they’ll need to use every time they cross the bridge. Person A can cross the bridge in 1 minute, Person B in 2 minutes, Person C in 5 minutes, and Person D in 8 minutes. When two people cross the bridge together, they must move at the slower person’s pace. Can all four get across the bridge in 15 minutes or less?

QUESTION 4: During a recent census, a man told a census taker that he had three children. When asked their ages, he replied, “The product of their ages is 72. The sum of their ages is the same as my house number.” The census taker ran to the man’s front door and looked at the house number. “I still can’t tell,” she complained. The man replied, “Oh, that’s right, I forgot to tell you that the oldest one likes chocolate pudding.” The census taker then promptly wrote down the ages of all three children. How old are they?

QUESTION 5: There are five bags of gold that all look identical, and each contains ten gold pieces. One of the five bags has fake gold, though. All five bags are identical, and the real gold and fake gold are identical in every way, except the pieces of fake gold each weigh 1.1 grams and the pieces of real gold each weigh 1 gram. You have a perfectly accurate digital scale available to you, but you can only use it once. How do you determine which bag has the fake gold?


[Image courtesy of AwesomeJelly.com.]

Okay, here’s your spoiler alert warning before we start unraveling these brain teasers.

So if you don’t want to see them, turn away now!

Last chance!

Ready? Okay, here we go!


[Image courtesy of Just Hourglasses.com.]

QUESTION 1: If you have a 7-minute hourglass and an 11-minute hourlass, how can you boil an egg for exactly 15 minutes?

A variation on the two jugs of water puzzle we’ve covered before, this puzzle is basically some simple math, though you need to be a little abstract with it.

  • Step 1: Start boiling the egg and flip over both hourglasses.
  • Step 2: When the 7-minute hourglass runs out, flip it over to start it again. (That’s 7 minutes boiling.)
  • Step 3: When the 11-minute hourglass runs out, the 7-minute hourglass has been running for 4 minutes. Flip it over again. (That’s 11 minutes boiling.)
  • Step 4: When the 7-minute hourglass runs out, another 4 minutes has passed, and you’ve got your 15 minutes of egg-boiling time.

QUESTION 2: Name the next number in the following sequence: 1, 11, 21, 1211, 111221, 312211, _____.

The answer is 13112221. This looks like a math or a pattern-matching puzzle, but it’s far more literal than that.

Each subsequent number describes the number before it. 11, for instance, isn’t eleven, it’s one one, meaning a single one, representing the number before it, 1.

The third number, 21, isn’t twenty-one, it’s “two one,” meaning the previous number consists of two ones, aka 11.

The fourth number, 1211, translates to “one two, one one,” or 21. The fifth number, 111221, becomes “one one, one two, two one.” And the sixth, 312211, becomes “three one, two two, one one.”

So, the number we supplied, 13112221, is “one three, one one, two two, two one.”


[Image courtesy of Do Puzzles.]

QUESTION 3: Four people want to cross a river, but the only option is a narrow bridge. The bridge can only support two people at a time. It’s nighttime, and the group has one torch, which they’ll need to use every time they cross the bridge. Person A can cross the bridge in 1 minute, Person B in 2 minutes, Person C in 5 minutes, and Person D in 8 minutes. When two people cross the bridge together, they must move at the slower person’s pace. Can all four get across the bridge in 15 minutes or less?

Yes, you can get all four across the bridge in 15 minutes.

This one’s a little tougher, because people have to cross the bridge in both directions so that the torch remains in play. Also, there’s that pesky Person D, who takes so long to get across.

So what’s the most time-efficient way to get Person D across? You’d think it would be so send D across with Person A, so that way, you lose 8 minutes with D, but only 1 minute going back with the torch with A. But that means only 6 minutes remain to get A, B, and C across. If you send A and C together, that’s 5 minutes across with C, and 1 minute back with A, and there’s your 15 minutes gone, and A and B aren’t across.

So the only logical conclusion is to send C and D across together. That’s 8 minutes down. But if you send C back down, that’s another 5 minutes gone, and there’s no time to bring A, B, and C back across in time.

So, C and D have to cross together, but someone faster has to bring the torch back. And suddenly, a plan comes together.

  • Step 1: A and B cross the bridge, which takes 2 minutes. A brings the torch back across in 1 minute. Total time used so far: 3 minutes.
  • Step 2: C and D cross the bridge, which takes 8 minutes. B brings the torch back across in 2 minutes. Total time used so far: 13 minutes.
  • Step 3: A and B cross the bridge again, which takes 2 minutes. Total time used: 15 minutes.

(It technically doesn’t matter if A returns first and B returns second or if B returns first and A returns second, so long as they are the two returning the torch.)


QUESTION 4: During a recent census, a man told a census taker that he had three children. When asked their ages, he replied, “The product of their ages is 72. The sum of their ages is the same as my house number.” The census taker ran to the man’s front door and looked at the house number. “I still can’t tell,” she complained. The man replied, “Oh, that’s right, I forgot to tell you that the oldest one likes chocolate pudding.” The census taker then promptly wrote down the ages of all three children. How old are they?

Their ages are 3, 3, and 8.

Let’s pull the relevant information from this puzzle to get started. There are three children, and the product of their ages is 72.

So let’s make a list of all the three-digit combinations that, when multiplied, equal 72: 1-1-72, 1-2-36, 1-3-24, 1-4-18, 1-6-12, 1-8-9, 2-2-18, 2-3-12, 2-4-9, 2-6-6, 3-3-8, 3-4-6. We can’t eliminate any of them, because we don’t know how old the man is, so his children could be any age.

But remember, after being told that the sum of the children’s ages is the same as the house number, the census taker looks at the man’s house number, and says, “I still can’t tell.” That tells us that the sum is important.

Let’s make a list of all the sums of those three-digit combinations: 74, 39, 28, 23, 19, 18, 22, 17, 15, 14, 14, 13.

The census taker doesn’t know their ages at this point. Which means that the sum has multiple possible combinations. After all, if there was only one combination that formed the same number as the house number, the census taker would know.

And there is only one sum that appears on our list more than once: 14.

So the two possible combinations are 2-6-6 and 3-3-8.

The chocolate pudding clue is the deciding fact. The oldest child likes chocolate pudding. Only 3-3-8 has an oldest child, so 3-3-8 is our answer.


[Image courtesy of Indy Props.com.]

QUESTION 5: There are five bags of gold that all look identical, and each contains ten gold pieces. One of the five bags has fake gold, though. All five bags are identical, and the real gold and fake gold are identical in every way, except the pieces of fake gold each weigh 1.1 grams and the pieces of real gold each weigh 1 gram. You have a perfectly accurate digital scale available to you, but you can only use it once. How do you determine which bag has the fake gold?

With only one chance to use the scale, you need to maximize how much information you can glean from the scale. That means you need a gold sample from at lesst four bags (because if they all turn out to have real gold, then the fifth must be fake). But, for the sake of argument, let’s pull samples from all five bags.

How do we do this? If we pull one coin from each bag, there’s no way to distinguish which bag has the fake gold. But we can use the variance in weight to our advantage. That .1 difference helps us.

Since all the real gold will only show up before the decimal point, picking a different number of coins from each bag will help us differentiate which bag has the fake gold, because the number after the decimal point will vary.

For instance, if you take 1 coin from the first bag, 2 coins from the second, 3 coins from the third, 4 coins from the fourth, and 5 coins from the fifth, you’re covered. If the fake gold is in the first bag, your scale’s reading will end in .1, because only one coin is off. If the fake gold is in the second bag, your scale’s reading will end in .2, because two coins are off. And so on.


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PN Trivia Scavenger Hunt: Answers & Winner!

[Image courtesy of Alaris Health.]

Thank you to everyone who entered our anniversary trivia scavenger hunt! Plenty of solvers, puzzlers, and PuzzleNationers tried their hand at answering all five questions before the deadline at midnight on Wednesday, and many succeeded!

Alas, there can be only one winner. But before we get to that, let’s look at the answers, shall we?


PuzzleNation Anniversary Trivia Scavenger Hunt

1.) One of my favorite recurring features is Puzzles in Pop Culture, where I explore puzzly moments in television, film, and literature. We’ve discussed Sherlock, Hell’s Kitchen, and even Gilmore Girls in installments of Puzzles in Pop Culture.

Question: How do you solve the four gallons of water puzzle?

Answer: There were actually two answers featured in the August 19, 2014 post “Puzzles in Pop Culture: Die Hard with a Vengeance” referenced in this question. Here’s the answer our winner submitted:

1. Fill the 3-gallon jug and pour the water into the 5-gallon jug.
2. Refill the 3-gallon jug and pour the water into the 5-gallon jug until the 5-gallon jug is full, leaving 1 gallon in the 3-gallon jug.
3. Empty the 5-gallon jug and pour the 1 gallon of water from the 3-gallon jug into the 5-gallon jug.
4. Fill the 3-gallon jug again and empty it into the 5-gallon jug, leaving exactly 4 gallons in the 5-gallon jug.


2.) You can’t talk about puzzles without also discussing games, because there’s so much overlap between the two. Game reviews from a puzzle solver’s perspective have become a part of the fabric of PuzzleNation Blog, as has creating your own puzzles and games from scratch.

Question: What’s the name of the DIY game that only requires a bunch of identical blank pieces of paper (like index cards) and something to write with?

Answer: Discussed in our September 15, 2015 post “DIY Pencil and Paper Puzzles,” this game is known as 1000 Blank White Cards.


3.) Naturally, if you’re going to talk puzzles, Sudoku is going to be part of the conversation sooner rather than later. We’ve not only explored the history of Sudoku here, but we’ve been a part of it, debuting brand-new Sudoku variants created by topnotch constructors.

Question: What do you call two overlapping Samurai Sudoku?

Answer: We posted many different Sudoku variants in our December 4, 2014 post “The Wide World of Sudoku,” but the puzzle in question is known as Shogun Sudoku.


4.) A fair amount of puzzle history, both past and present, has been covered here over the last five years. We’ve examined cryptography in the American Revolution, the Civil War, both World Wars, and beyond. We’ve celebrated the one-hundredth anniversary of the crossword. And we’ve even discussed scandals in the puzzle world.

Question: What are the names of the programmer and crossword constructor who first uncovered the curious pattern of puzzle repetition in USA Today and Universal Uclick puzzles that eventually led to the ouster of Timothy Parker?

Answer: As discussed in a series of posts entitled “Puzzle Plagiarism,” the programmer’s name is Saul Pwanson and the constructor’s name is Ben Tausig.


5.) In the Internet age, memes and fads appear and disappear faster than ever. A picture or a joke or a news story can sweep the world in a matter of hours, and then vanish forever. On a few occasions, the Internet has become obsessed with certain optical illusions, and we’ve done our best to analyze them from a puzzler’s perspective.

Question: The creators of The Dress appeared on what talk show to put the mystery to bed once and for all?

Answer: Discussed on March 6, 2015 in a Follow-Up Friday post, the mystery of The Dress was laid to rest on The Ellen DeGeneres Show.


[Image courtesy of ClipArt Panda.]

And now, without any further ado, we’d like to congratulate our winner, who shall remain nameless. After all, like a lottery winner, she doesn’t want to be mobbed by those hoping for a piece of the action. =)

She’ll be receiving her choice of either a Penny Dell Crosswords App puzzle set download OR a copy of one of the puzzle games we’ve reviewed this year!

Thank again to everyone for playing and for celebrating five years of PuzzleNation Blog with us. We truly could not have done it without you!


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Get Into the Puzzly Swing of Things with Swinger!

Puzzles are infinitely adaptable. They’ve gone from riddles to pen and paper puzzles to mechanical puzzles made of wood, steel, plastic, and onward into video games, and now, apps on your phone. More puzzle apps than ever before are available on all platforms; it’s the new frontier of puzzle-games.

One of the puzzle apps that has recently been making some waves in the online market is Swinger, an addictive little puzzly delight.

In Swinger, your goal is to capture the little blue sparkling objects known as balls. To do so, you must tag them with a swinging arm that moves back and forth at your command.

Tap on one of those green rings to anchor one end of your swinger, and it will circle that ring like the spoke of a bicycle wheel. You can tap that same green ring to change direction, swinging your swinger back and forth as needed.

Navigate the ever-changing landscape of green rings to capture the balls, but be careful to avoid the obstacles that crop up over time, because they’ll block your swinger and damage your overall score.

A fun mix of quick reflexes and clever strategy, Swinger is all about adapting to your environment. Timing is key as your swinger swings toward each blue ball, before you tap another ring to move it, either to avoid one of the obstacles or to snag the next ball on the screen. You might even find yourself flipping the grid around in order to jump your swinger from one ring to another!

The concept is simple, but the gameplay grows increasingly complex with each passing screen as the obstacles become more bothersome and the swinger varies in size from ring to ring.

It’s an interesting mechanic that I haven’t seen in app form before. (For long-time video game fans, it has the feel of advanced platform games like Bionic Commando, Ratchet & Clank, or Super Mario Galaxy!)

Swinger is available from PiPiPass Studios on Google Play for Android devices.


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Puzzle History: Codebreaking and the NSA, part 3

[Image courtesy of NSA’s official Twitter account.]

At the end of part 2 in our series, we left off during the early days of the NSA, as American cryptographers continued to labor under the shadow of the Black Friday change in Russian codes.

You may have noticed that part 2 got a little farther from puzzly topics than part 1, and there’s a reason for that. As the NSA evolved and grew, codebreaking was downplayed in favor of data acquisition. The reasons for this were twofold:

1. Context. You need to understand why given encrypted information is important in order to put it toward the best possible use. As Budiansky stated in part 1, “The top translators at Bletchley were intelligence officers first, who sifted myriad pieces to
assemble an insightful whole.”

2. Russian surveillance and bugging continued to grow more clever and sophisticated, pushing attention away from codebreaking. After all, what good is breaking codes or developing new ones if they can just steal unencrypted intel firsthand by monitoring
agents in the field?

Moving forward, the NSA would continue to pursue all manner of data mining, eventually leaving behind much of the codebreaking and analysis that originally formed the backbone of the organization. But that was in years to come. Cryptography was still a major player in NSA operations from the ’50s and onward.

[The progression of “secret” and “top secret” code words.
Image courtesy of NSA’s official Twitter account.]

In May 1956, NSA cryptanalytic veterans pushed a proposal titled “Recommendations for a Full-Scale Attack on the Russian High-Level Systems,” believing that specially designed computers from IBM could provide the key for cracking the impenetrable Russian cryptography wall. Some cryptographers believed that ever-increasing processor speeds would eventually outpace even sophisticated codes.

By 1960, the NSA had spent $100 million on computers and analytical tools.

The problem? The NSA was collecting so much information that their increasingly small team of cryptoanalysts couldn’t dream of processing even a tiny portion of it.

But the quest for data access would only grow more ambitious.

In the wake of Sputnik’s launch in October of 1957, US signals intelligence would go where no man had gone before. The satellite GRAB, launched alongside Transit II-A in June of 1960, was supposedly meant to study cosmic radiation. (GRAB stood for Galactic Radiation and Background.)

[Image courtesy of NSA’s official Twitter account.]

But it was actually intended to collect radar signals from two Soviet air-defense systems. This was the next step of ELINT, electronic intelligence work. (The younger brother of SIGINT.)

The NSA would later find a huge supporter in President Lyndon Johnson, as the president was heavily invested in SIGINT, ELINT, and any other INTs he could access. This did little to quell the intelligence-gathering rivalry growing between the CIA and NSA.

Of course, that’s not to say that the NSA ceased to do any worthwhile work in codebreaking. Far from it, actually.

During the Vietnam War, NSA analysts pored over North Vietnamese signals, trying to uncover how enemy pilots managed to scramble and respond so quickly to many of the US’s airstrikes conducted during Operation Rolling Thunder.

Careful analysis revealed an aberrant character (in Morse code) in messages that appeared in North Vietnamese transmissions before 90 percent of the Rolling Thunder airstrikes. By identifying when the enemy used that aberrant character, the analysts
were able to warn US pilots whether they were heading toward a prepared enemy or an unsuspecting one during a given sortie.

Other NSA teams worked to protect US communications by playing the role of an enemy analyst. They would try to break US message encryptions and see how much they could learn from intercepted US signals. Identifying flaws in their own procedures — as well as members of the military who were cutting corners when it came to secured communications — helped to make US communications more secure.

[Image courtesy of NSA.gov.]

In 1979, Jack Gurin, the NSA’s Chief of Language Research, wrote an article in the NSA’s in-house publication Cryptolog, entitled “Let’s Not Forget Our Cryptologic Mission.” He believed much of the work done at the agency, and many of the people
hired, had strayed from the organization’s core mission.

The continued push for data acquisition over codebreaking analysis in the NSA led to other organizations picking up the slack. The FBI used (and continues to use) codebreakers and forensic accountants when dealing with encrypted logs from criminal organizations covering up money laundering, embezzlement, and other illegal activities.

And groups outside the government also made impressive gains in the field of encryption, among them IBM’s Thomas J. Watson Research Center, the Center for International Security and Arms Control, and even graduate student programs at universities like MIT and Stanford.

For instance, cryptographer Whitfield Diffie developed the concept of the asymmetric cipher. Joichi Ito explains it well in Whiplash:

Unlike any previously known code, asymmetric ciphers do not require the sender and receiver to have the same key. Instead, the sender (Alice) gives her public key to Bob, and Bob uses it to encrypt a message to Alice. She decrypts it using her private key. It no longer matters if Eve (who’s eavesdropping on their conversation) also has Alice’s public key, because the only thing she’ll be able to do with it is encrypt a message that only Alice can read.

This would lead to a team at MIT developing RSA, a technique that implemented Diffie’s asymmetric cipher concept. (It’s worth noting that RSA encryption is still used to this day.)

[Image courtesy of Campus Safety Magazine.com.]

The last big sea change in encryption came when the government and military realized they no longer had a monopoly on codebreaking technology. Increased reliance and awareness of the importance of computer programming, greater access to computers with impressive processing power, and a groundswell of support for privacy from prying government eyes, led to dual arms races: encryption and acquisition.

And this brings us to the modern day. The revelations wrought by Edward Snowden’s leak of NSA information revealed the incredible depth of government data mining and acquistion, leading some pundits to claim that the NSA is “the only part of government that actually listens.”

Whatever your feelings on Snowden’s actions or government surveillance, there is no doubt that the National Security Agency has grown and changed a great deal since the days of cracking the ENIGMA code or working with the crew at Bletchley Park.

Where will American codebreaking go next? Who knows? Perhaps quantum computing will bring codes so complicated they’ll be impenetrable.

All I know is… it’s part of puzzle history.


I hope you enjoyed this multi-part series on the history of 20th-century codebreaking in America. If you’d like to learn more, you can check out some of the valuable sources I consulted while working on these posts:

Code Warriors: NSA’s Codebreakers and the Secret Intelligence War Against the Soviet Union by Stephen Budiansky

Whiplash: How to Survive Our Faster Future by Joichi Ito

The Secret Lives of Codebreakers by Sinclair McKay


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Puzzle History: Codebreaking and the NSA, part 2

[Image courtesy of NSA’s official Twitter account.]

At the end of part 1 of our look at the history of the NSA and American codebreaking, we left off with the pivotal Black Friday event.

On November 1, 1948, all intel coming from monitored Soviet signals went quiet. All traffic on military, naval, and police radio links was replaced with dummy messages. It was such an unprecedented and alarming event that London and Washington briefly considered that it might’ve been the first indication of preparations for war.

According to Code Warriors author Stephen Budiansky:

The full extent of the disaster only became apparent the following spring when real traffic started reappearing on the radio nets, now employing greatly improved — and completely unbreakable — technical and security procedures. The keying errors or other mistakes that had allowed most of the Soviets’ machine-enciphered military traffic to be routinely read by US and British codebreakers for the last several years had been corrected, and the much more disciplined systems that now replaced them slammed the cryptanalytic door shut.

Even the one-time pads that had offered some hope to attentive American codebreakers were updated, eliminating the ability to sort messages by which organization they originated from.

Codemakers had suddenly outpaced codebreakers.

[The Kryptos sculpture outside CIA Headquarters. The NSA cracked
several of its codes before the CIA did. Image courtesy of Slate.com.]

The Office of Naval Intelligence wanted to take over from Signals Intelligence (SIGINT), demanding to see “everything” so they could do the job. They claimed SIGINT should limit their work to message translation, leaving interpretation to “the real experts.” This sort of territorial gamesmanship would continue to hamper government organizations for decades to come.

And that demand to see everything? That probably sounds familiar, in light of the revelations about government data collection and the PRISM program that were revealed in Edward Snowden’s leaks.

Black Friday was the start of all that, a shift from codecracking to the massive data collection and sifting operation that characterized the NSA for decades to come.

More amazingly, there was SO MUCH information collected during World War II that SIGINT was still poring over it all in 1949, decrypting what they could to reveal Soviet agents in the U.S. and England.

The fact that a high-ranking member of British Intelligence at the time, Kim Philby, was actually a Soviet double agent complicated things. After a decade under suspicion, Philby would flee to the Soviet Union in 1963, stunning many friends and colleagues who had believed in his innocence.

[The spy and defector, honored with a Soviet stamp.
Image courtesy of Britannica.com.]

Although the Russians had flummoxed SIGINT, other countries weren’t so lucky. The East German police continued to use ENIGMA codes as late as 1956. Many of the early successes in the Korean War were tied to important decryption and analysis work by SIGINT. Those successes slowed in July of 1951, when North Korea began mimicking Russia’s radio procedures, making it much harder to gain access to North Korean intel.

Finally, the chaotic scramble for control over signal-based data gathering and codebreaking between the government and the military resulted in the birth of the National Security Agency on November 4, 1952, by order of President Truman.

One of the first things the NSA did? Reclassify all sorts of material involving historical codebreaking, including books and papers dating back to the Civil War and even the American Revolution.

[The actual report that recommended the creation of the NSA.
Image courtesy of NSA’s official Twitter account.]

The creation of the NSA had finally, for a time at least, settled the issue of who was running the codebreaking and signals intelligence operation for the United States. And they were doing fine work refining the art of encryption, thanks to the work of minds like mathematician and cryptographer Claude Shannon.

One of Shannon’s insights was the inherent redundancy that is built into written language. Think of the rules of spelling, of syntax, of logical sentence progression. Those rules define the ways that letters are combined to form words (and those words form sentences, and those sentences form paragraphs, and so on).

The result? Well, if you know the end goal of the encoded string of characters is a functioning sentence in a given language, that helps narrow down the amount of possible information contained in that string. For instance, a pair of characters can’t be ANYTHING, because letter combinations like TD, ED, LY, OU, and ING are common, while combos like XR, QA, and BG are rare or impossible.

By programming codecracking computers to recognize some of these rules, analysts were developing the next generation of codebreakers.

Unfortunately, the Russian line was holding. The NSA’s failure to read much, if any, Soviet encrypted traffic since Black Friday was obviously becoming more than just a temporary setback.

Something fundamental had changed in the nature of the Russian cryptographic systems, and in the eyes of some scientific experts called in to assess the situation, the NSA had failed to keep up with the times.


I hope you’re enjoying this look at the early days of America’s 20th-century codebreaking efforts. Part 3 will continue next week, with the sea change from active codebreaking to data mining, plus Vietnam, the space race, and more!


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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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