Solution to Riddle of the Week

Solution to Riddle of the Week: The Hardest Logic Puzzle Ever

Difficulty level: Hardest Ever

By Jay Bennett
MICHAEL STILLWELL

This puzzle is a variation of the knights and knaves puzzles created by American mathematician Raymond Smullyan. It was published for the first time in its basic form by American philosopher and logician George Boolos, who included it in the Harvard Review of Philosophy in 1996. American computer scientist John McCarthy is credited with adding the challenge of not knowing the meanings of ja and da.

Riddle of the Week

Riddle of the Week: The Hardest Logic Puzzle Ever

Difficulty level: Hardest Ever

By Jay Bennett
MICHAEL STILLWELL

Sailing through a thick fog, you come upon a mysterious island shrouded in mist. A towering volcano in the center of the island pierces the clouds, billowing smoke into the sky. You land your boat and set out to ascend the peak. After an arduous climb, you approach the volcano summit, where lava glows red within a vast crater.

Here, you are approached by three gods.

Solution to Riddle of the Week

Solution to Riddle of the Week: Edgar Allan Poe's Riddle Poem

Difficulty level: Hard

By Jay Bennett

 MICHAEL STILLWELL

Some of the literary figures in Poe's poem Enigma are much more obvious than others, partially due to the relevance that these writers have maintained in the 21st century. Some are from the ancient world, and some are Poe's contemporaries.

Many of the storytellers that Poe identifies had a knack for penning speculative poems—tales of dreams and faeries and monsters and gods and the like. In a way, the poem Enigma is Poe's list of the founders of fantasy, science fiction, and horror.

Riddle of the Week

Riddle of the Week: Edgar Allan Poe's Riddle Poem

Difficulty level: Hard

By Jay Bennett

 MICHAEL STILLWELL

Problem

The following poem was published on February 2, 1833, in the Baltimore Saturday Visiter. It contains descriptions and clues of 11 famous literary figures. The poem was only attributed to "P." However, 20th century literature professor Thomas Ollive Mabbott credits Edgar Allan Poe with writing the poem. Mabbott also managed to identify all 11 literary figures hidden in the verse.

Can you?

Riddle of the Week

Riddle of the Week: Knights and Knaves, 

Part 2

Difficulty level: Easy

MICHAEL STILLWELL

By Jay Bennett

Welcome back to a series of Knights and Knaves logic puzzles. These riddles take place on an island where there are two types of people, knights, who always tell the truth, and knaves, who always lie. You can see the first knights and knaves puzzle here.

Problem

On the island of knights and knaves, you are approached by two people. The first one points to the second and says, "he is a knave." The second one then says, "neither of us are knaves."

What are they actually?

Hint

On many of these puzzles, especially the harder ones going forward, it will be important to label everyone. So A says, "B is a knave." And B says, "neither A nor B are knaves."

Solution to Riddle of the Week

Solution to Riddle of the Week: Knights and Knaves, Part 1

Difficulty level: Easy

Michael Stillwell

By Jay Bennett

The first person you meet on this island is not to be trusted.

Solution

The first person cannot be a knight. Knights always tell the truth, so if he were a knight, he would tell you so. Instead he says, "we are both knaves."

If the first man is a knave, then he must be a liar. This means they cannot both be knaves. So, since we've established that the first man is a knave and a liar, the second man must be a knight.

Easy? Well this island of knights and knaves is only going to get stranger and more difficult to navigate, so make sure to come back next week for another riddle!

Riddle of the Week

Riddle of the Week: Knights and Knaves, Part 1

Difficulty level: Easy

Michael Stillwell

By Jay Bennett

Ready for a riddle series? In the coming weeks, Popular Mechanics will present progressively harder "knights and knaves" puzzles. These logic puzzles take place on an island with two types of people: the knights, who always tell the truth, and the knaves, who always lie.

American mathematician and musician Raymond Smullyan named this type of puzzle in his 1978 book What Is the Name of This Book? You might remember a knights and knaves puzzle from the 1986 fantasy film, Labyrinth.

To start the series off, let's have a simple conversation with these knaves and these knights.

Problem

On the island of knights and knaves, you are approached by two people. The first one says to you, "we are both knaves."

What are they actually?

Hint

Is the first person a knight?

Solution

When you know if these two fellows are knights or knaves, or one of each, check back on Friday for the solution.

Solution to Riddle of the Week

Solution to Riddle of the Week: Can You Find the Hiding Turkey?

Difficulty level: Hard

Betsy Farrell
By Jay Bennett

Tracking the tricky turkey may seem impossible, as it seems the bird could stay one step ahead. However, there is a way to make sure you find that felonious fowl.

Solution

Randomly searching will not guarantee you find the turkey. Neither will checking every box, nor will checking the same box over and over. To find this bird, we are going to have to make some assumptions.

First, for the sake of argument, let's assume the turkey is in an even-numbered box, meaning either box 2 or box 4.


Betsy Farrell

Let's say you check box 2. If you find the turkey, all is well and good in the world and Thanksgiving can proceed. If not, then you know the turkey must have been in box 4 (again, this is based on an initial assumption that the turkey was in an even-numbered box).

If the turkey was in box 4 on the first day, when you checked box 2, then it must move to either box 3 or box 5 on the second day. So on the second day, check box 3. If the turkey is there, you win. If not, it must be in box 5, and if the turkey is in box 5 on the second day, it must move to box 4 on the third day, and so you check box 4 on the third day and find the turkey.

Now, the above scenario—checking box 2, 3, and then 4—will always let you win assuming the turkey started in an even-numbered box. But, of course, that might not be the case. Now let's look at the scenario if the turkey started in an odd-numbered box—1, 3, or 5.


Betsy Farrell

If the turkey is in box 1, 3, or 5, then on the second day, it must have moved to either box 2 or 4. On the third day, it must have moved back to box 1, 3, or 5. And on the fourth, the turkey again must have moved to either box 2 or 4.

You can probably sense we've discovered something important here: If the turkey started in an odd-numbered box, then after checking for three days, it must be in an even-numbered box. In other words, if the turkey started in an odd-numbered box, at the start of the fourth day, it must be in an even-numbered box. We now must combine the two scenarios.

First, we know from the first example that if you check box 2 and then 3 and then 4, you will find the turkey if it started in an even-numbered box. Let's say you check 2, 3, and 4 on the first three days, and you do not find the turkey. That means it must have started in an odd-numbered box, which also means that on the start of the fourth day, it must be in an even-numbered box. So, on the fourth day, if you have not found the turkey, you repeat the process, because you know that now it must be in an even box.

So here is the solution: Check box 2 on the first day, then 3 on the second day, and then 4 on the third day. If the turkey was in an even box, you are guaranteed to find it on one of those first three days. If you don't find it, then it must have started in an odd-numbered box, and on the start of the fourth day, it must be in an even-numbered box. So you then check box 2 on the fourth day, then box 3 on the fifth day, and finally box 4 on the sixth day. No matter what, you will have found the turkey.

In short: check box 2 then 3 then 4, and if you do not find the turkey, check box 2 then 3 then 4 again.

Riddle of the Week

Riddle of the Week: Can You Find the Hiding Turkey?

Difficulty level: Hard

Betsy Farrell
By Jay Bennett

Problem

A week before Thanksgiving, a sly turkey is hiding from a family that wants to cook it for the holiday dinner. There are five boxes in a row, and the turkey hides in one of these boxes. Each night, the turkey moves one box to the left or right, hiding in an adjacent box the next day. Each morning, the family can look in one box to try to find the turkey.

How can the family guarantee they will find the turkey before Thanksgiving dinner?

Hint


Betsy Farrell

If you simply search one box per day, starting with 1 and finishing with 5, you might not find the turkey. It could be in box 3 when you check box 2, and then move to 2 the next day when you check 3, and you would miss it. Similarly, you cannot keep checking the same box, as the turkey could bounce back and forth between only two boxes the entire time. You need a plan to guarantee you will find the turkey.

Solution

Once you help this family find their Thanksgiving dinner, you can check back for the solution on Friday.

Solution to Riddle of the Week

Solution to Riddle of the Week: The Entrance to the Thieves Guild

Difficulty level: Easy

Michael Stillwell

By Jay Bennett

It seems the businessman is not as clever as he thought. What is another way the numbers that the guard asked the first two people could produce the numbers they gave in reply? You can check the original question here.
Solution

The businessman should have said "three," the number of letters in "ten." Similarly, "twelve" has six letters, and "six" has three letters, the answers the first two thieves gave.

It seems the businessman will have to look elsewhere to fence his stolen goods

Riddle of the Week

Riddle of the Week: The Entrance to the Thieves Guild

Difficulty level: Easy

Michael Stillwell

By Jay Bennett

Welcome back to the weekly riddle series from Popular Mechanics. Today we try to help a naive fellow gain access to a shadowy guild of thieves.

Problem

A businessman, for personal reasons of his own, needs to gain access to a notorious but elusive thieves guild. After snooping around the city for a few days, he finally gets a tip about the location of the guild entrance down a dark alleyway.

After observing the entrance for about an hour, the businessman notices that the thieves seem to have some sort of passcode system using numbers. When someone approaches the door, they are given a number, and then they reply with a number.

The first person steps up, and the guard tells them "twelve," to which they respond "six." The person is admitted. A second person approaches the door, and the guard tells them "six," to which they reply "three." The second person is also admitted.

Convinced that he has cracked the code, the businessman approaches the door to the thieves guild, and the guard tells him "ten," to which he confidently replies "five." The guard immediately slams the door in the businessman's face, and a deadbolt slots into place.

What should the businessman have said?

Hint

No hint this week.

Solution to Riddle of the Week

Solution to Riddle of the Week: A Shipment of Apples to Bananaville

Difficulty level: Hard

Michael Stillwell
 
By Jay Bennett

Obviously you cannot load up your truck with 1,000 apples and drive directly to Bananaville 1,000 miles away. All of your apples would go to the apple toll. You must devise a way to bring some apples part of the way, go back and get more, then get a full truckload to start with closer to Bananaville. You can see the original problem here.

SOLUTION

You can deliver a total of 833 apples to Bananaville. Let's see how.

First you depart Appleland with a full truckload of 1,000 apples and drive 333 miles, about a third of the way to Bananaville, and drop off the 667 apples you have left after tolls. Then you go back and repeat this process twice, which leaves you with 2,001 apples at the 333-mile mark.

The second leg of the journey is similar, but this time you will take two loads of 1,000 apples 500 miles further, which will give you 1,000 apples at the 833-mile mark with 167 miles still to go to Bananaville. (Alas, you must abandon one apple behind, for it will not fit into your truck.)

Finally, take the 1,000 apples you have left and drive the remaining 167 miles to Bananaville, arriving with 833 apples. The citizens of the noble town will surely thank you for adding a little variety to their diet.

You may have noticed a mathematical quirk to the solution. In the first step, you took three truckloads a third of the way (333 miles). But in the second step, you took two truckloads half of the overall distance (500 miles). This is the most efficient way to transport the apples because you leave yourself with the correct remaining number of apples for full truckloads, meaning products of 1,000. Because 1,000 miles doesn't cleanly divide by 3, you are forced to leave that one apple behind at the 333-mile mark of your journey.

It may have taken a bit more time than otherwise, but you have successfully brought the apple trade to Bananaville, and you paid as little as possible in apple tolls as well! Good work.

Riddle of the Week

Riddle of the Week: A Shipment of Apples to Bananaville

Difficulty level: Hard

Michael Stillwell
 
By Jay Bennett

Welcome back. This week we try to help the fictional town of Bananaville, which, despite its predilection for the yellow fruit, is eagerly awaiting a shipment of delicious apples.

PROBLEM

You are tasked with transporting 3,000 apples from Appleland to Bananaville, a distance of 1,000 miles. You have a truck that holds 1,000 apples. However, there is an apple toll on the road to Bananaville, and you must pay 1 apple per mile you drive. There is no toll when you are headed in the opposite direction, toward Appleland.

What is the largest number of apples you can transport to Bananaville?

HINT

You may leave apples on the side of the road and return to pick them up later.

SOLUTION

Once you bring as many apples as you possibly can to the potassium-enriched citizens of Bananaville, you can see the solution on Friday.

Solution to Riddle of the Week

Solution to Riddle of the Week: Getting the Goat

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

What's interesting about this puzzle is that there is an infinite number of possibilities for the number of cows that the brothers start with, and yet the amount of money owed the brother who gets the goat will always be the same. You can revisit the original question here.

SOLUTION

The brother who gets all lambs must pay $2 to the brother who gets the goat to equalize the share of the profits from the cow sales.

Let's take a closer look at this. If we assume that the brothers started with 10 cows, then they sold those cows for 10 dollars each (because each cow is sold for as many dollars as the brothers have cows), netting the two brothers $100. Then they would have enough money to buy 10 lambs for $10 apiece. But in this case, the brothers have no money left over to buy a kid goat, which the riddle tells us they do. So the brothers could not have started with 10 cows. This examples tells us something us: They could not have started with a number of cows that would sell for a dollar amount that is cleanly divisible by 10, or else they would have no money left over for the goat.

Here's another piece of information that proves crucial to solving the riddle: The amount of money the brothers receive for selling the cows must be a perfect square. Because each cow is sold for as many dollars as the brothers have cows, the amount of money they receive will always be a number multiplied by itself.

Let's try another number for starting cows, say, 15. In this scenario, the brothers sell each cow for $15 dollars, netting them a profit of $225. They can then buy 22 lambs for $10 apiece, leaving $5 left over to buy the lamb. However, this example would give the brothers 22 lambs and one goat. That's a total of 23 animals, and the question tells you the brothers end up with an even number of animals, so 15 cannot be the right number of starting cows.

This failed example gives us the final piece of information we need to solve the riddle. Because the final number of animals must be even, we know the brothers must have an amount of money that allows them to buy an odd number of lambs. That way, once you add in the goat, you've got an even number. In other words, the digit in the tens place of the amount of money they receive from selling cows must be odd. We already know that amount of money will be a square number, so let's take a look at the first 20 square numbers.



We have discovered two key pieces of information: The amount of money received from cow sales cannot have a factor of 10, and it must have an odd number in the tens place. Assuming the brothers have between 1 and 20 cows to start, this conditions are met for 4 cows, 6 cows, 14 cows, and 16 cows.

Let's assume the brothers have 14 cows, so they sell them for $196. This would give them 19 lambs and a goat that cost $6. One brother gets 10 lambs worth $100, the other gets 9 lambs and a goat collectively worth $96. The first brother must pay the second brother $2 so the total transaction has an equal value of $98 per brother.

But here's the thing: If you plug any of the numbers in that work for the riddle, you will find that the goat must have cost $6 (the number in the ones place). If the goat cost $6, then the brother who receives it is owed half the difference of the cost of a lamb and the cost of the goat, or: 1/2 ($10 - $6) = $2. This is the answer to the riddle, the brother who gets the goat will always be owed $2.

If you look at a full list of square numbers, it will quickly become apparent that all numbers that work for the starting number of cows are the ones with 4 or 6 in the ones place. The brothers could have started with 16 cows, or 2,116 cows, and the one will still owe the other $2 when all is said and done.

It helps to practice a little math if you are going to get into dairy farming.

Riddle of the Week

Riddle of the Week: Getting the Goat

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

Welcome back to Popular Mechanics' Riddle of the Week! This week we take a look at a math riddle found in L.H. Longley-Cook's book Work This One Out: 105 Puzzling Brain-Teasers. A scholar of credibility theory, or the use of recent past events to change statistical predictions of future events, Longley-Cook also enjoyed penning math puzzles. This one is about two brothers who decide to divvy up the family dairy farm.

PROBLEM

Two brothers each have an equal stake in a dairy farm, but after the price of milk drops, they decide to go their separate ways and start raising lambs instead. The brothers sell off their cows, and sell each cow for as many dollars as they have cows. They use that money to buy lambs for $10 apiece. Once they have purchased as many lambs as they can with the money from the cow sales, they have a little left over, which they use to buy a kid goat.

The two brothers now have purchased an even number of animals. They split the animals evenly, but the brother who gets the goat wants a sum of money from the brother who gets all lambs to equalize the value.

How much money should the brother with the goat receive?

HINT

With math riddles like these, it is often best to start plugging in numbers and see what you get. Choose a number for the cows that the brothers have at the beginning and see if the rest of the parameters for the riddle can be satisfied. After a bit of trial and error, a pattern should emerge.

SOLUTION

Once you settle this financial dispute between the two livestock-raising brothers, you can check back for the solution on Friday.

Solution to Riddle of the Week

Solution to Riddle of the Week: The Dead Man in a Scuba Suit

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

These situational riddles can be tricky to logically explain, but once the solution is known the scenario suddenly becomes plausible. A similar riddle, with more of a clue to decipher, is The Bicycle Killer.

SOLUTION

The section of forest with the dead scuba diver was on fire. A water bomber aircraft scooped up water from a nearby ocean, and inadvertently snatched up the diver as well. The water put out the fire, but the scuba diver died from the fall from the aircraft.

Sometimes it's just your time.

Riddle of the Week

Riddle of the Week: The Dead Man in a Scuba Suit

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

This week we take a look at a curious situation and search for a logical explanation. The riddle can be summed up in one question: How the hell did this guy get here?

PROBLEM

There is a dead man in a scuba suit in the middle of a forest. How did he get there?

HINT

Generally, when you answer this riddle, you are permitted to ask yes or no questions. In this case, we will just provide you with a hint here:

The area of the forest with the diver is filled with blackened and charred trees and stumps.

Solution to Riddle of the Week

Solution to Riddle of the Week: A Sooty Train Ride

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

Like many a good riddle, you need to put yourself in the shoes of the characters. What would you be thinking if you were Miss Atkinson? What would you be thinking if you were Miss Atkinson thinking about what the other two passengers were thinking?

SOLUTION

Miss Atkinson knows that she has soot on her face because neither of the other two passengers stops laughing, or realizes their faces are sooty. If only two people had sooty faces, then they would see everyone laughing, one sooty face, and one clean face. Because the others are also laughing, they would know that their own face was sooty—if only two people had mucky faces.

Since neither of the other two realize their own face is dirty, Miss Atkinson realizes that her own must be as well, which would allow the other two passengers to believe their own faces clean and continue laughing.

Make sure to give this one to your fellow train car companions down the line, and make sure to come back for another riddle!

Riddle of the Week

Riddle of the Week: A Sooty Train Ride

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

Welcome back! We took a small hiatus from Popular Mechanics' weekly-ish series of riddles, but we are going to get back on the riddle train this week with a train riddle. Actually this week's riddle is more about dirty faces. The problem appears in the book Puzzle-Math by George Gamow, and it is credited to revolutionary Soviet astrophysicist Victor Ambartsumian.

PROBLEM

Three passengers are sitting in a train car with the window open as they travel cross-country. Suddenly, a freight locomotive laden with coal speeds by, knocking a cloud of soot straight into the passenger car's window. All three passengers' faces are covered in soot, but it happens so rapidly that none realize their own face is covered. When Miss Atkinson looks up from her book, she begins to laugh at how absurd the other two passengers look, who are also laughing.

Each passenger assumes that their own face is clean, and that the other two are laughing at each other. Then Miss Atkinson suddenly stops laughing and takes a handkerchief out of her purse to wipe her face clean.

How did she know she also had soot on her face?

HINT

In this puzzle, we can assume that all of the passengers are making logical assumptions, but Miss Atkinson is a bit quicker than her carriage companions.

Check back Friday for the Solution.

Solution to Riddle of the Week

Solution to Riddle of the Week: Le Havre to New York

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

It can be tricky to mentally visualize all the ship crossings, so grab that pen and pad. And remember, the correct answer includes all the ships that are already at sea when the target ship sets sail. You can see the original question here.

SOLUTION

The answer is 15 ocean liners. When one sets sail from Le Havre, there are 6 other ocean liners already at sea that set out from New York. In addition to these, 7 more ocean liners will leave New York during the voyage, including the ship that leaves simultaneously from New York as our ship casts off from Le Havre. So a ship leaving Le Havre "today" will see 13 other ships in the ocean. In addition to these, it will cross paths with another ship as it casts off from Le Havre, and yet another as soon as it arrives in New York, adding 2 more. An ocean liner leaving Le Havre on any given day will pass 15 other ocean liners on its way to New York.

If you have a hard time picturing it, check out the diagram below. The chart imagines a month of 30 days, and each line represents an ocean liner. Blue lines are casting off from Le Havre, and red lines are casting off from New York. If we suppose that the ship leaving "today" is leaving on the 7th of the month, you can see that it encounters 15 other ocean liners during its 7-day journey.


Michael Stillwell