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10 Paradoxes That Will Boggle Your Mind

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A paradox is a statement or problem that either appears to produce two entirely contradictory (yet possible) outcomes, or provides proof for something that goes against what we intuitively expect. Paradoxes have been a central part of philosophical thinking for centuries, and are always ready to challenge our interpretation of otherwise simple situations, turning what we might think to be true on its head and presenting us with provably plausible situations that are in fact just as provably impossible. Confused? You should be.

1. ACHILLES AND THE TORTOISE

The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m. When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise always moving forwards while Achilles always plays catch up.

Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he can overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.

2. THE BOOTSTRAP PARADOX

The Bootstrap Paradox is a paradox of time travel that questions how something that is taken from the future and placed in the past could ever come into being in the first place. It’s a common trope used by science fiction writers and has inspired plotlines in everything from Doctor Who to the Bill and Ted movies, but one of the most memorable and straightforward examples—by Professor David Toomey of the University of Massachusetts and used in his book The New Time Travellers—involves an author and his manuscript.

Imagine that a time traveller buys a copy of Hamlet from a bookstore, travels back in time to Elizabethan London, and hands the book to Shakespeare, who then copies it out and claims it as his own work. Over the centuries that follow, Hamlet is reprinted and reproduced countless times until finally a copy of it ends up back in the same original bookstore, where the time traveller finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?

3. THE BOY OR GIRL PARADOX

Imagine that a family has two children, one of whom we know to be a boy. What then is the probability that the other child is a boy? The obvious answer is to say that the probability is 1/2—after all, the other child can only be either a boy or a girl, and the chances of a baby being born a boy or a girl are (essentially) equal. In a two-child family, however, there are actually four possible combinations of children: two boys (MM), two girls (FF), an older boy and a younger girl (MF), and an older girl and a younger boy (FM). We already know that one of the children is a boy, meaning we can eliminate the combination FF, but that leaves us with three equally possible combinations of children in which at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child is a boy—MM—must be 1/3, not 1/2.

4. THE CARD PARADOX

Imagine you’re holding a postcard in your hand, on one side of which is written, “The statement on the other side of this card is true.” We’ll call that Statement A. Turn the card over, and the opposite side reads, “The statement on the other side of this card is false” (Statement B). Trying to assign any truth to either Statement A or B, however, leads to a paradox: if A is true then B must be as well, but for B to be true, A has to be false. Oppositely, if A is false then B must be false too, which must ultimately make A true.

Invented by the British logician Philip Jourdain in the early 1900s, the Card Paradox is a simple variation of what is known as a “liar paradox,” in which assigning truth values to statements that purport to be either true or false produces a contradiction. An even more complicated variation of a liar paradox is the next entry on our list.

5. THE CROCODILE PARADOX

A crocodile snatches a young boy from a riverbank. His mother pleads with the crocodile to return him, to which the crocodile replies that he will only return the boy safely if the mother can guess correctly whether or not he will indeed return the boy. There is no problem if the mother guesses that the crocodile will return him—if she is right, he is returned; if she is wrong, the crocodile keeps him. If she answers that the crocodile will not return him, however, we end up with a paradox: if she is right and the crocodile never intended to return her child, then the crocodile has to return him, but in doing so breaks his word and contradicts the mother’s answer. On the other hand, if she is wrong and the crocodile actually did intend to return the boy, the crocodile must then keep him even though he intended not to, thereby also breaking his word.

The Crocodile Paradox is such an ancient and enduring logic problem that in the Middle Ages the word "crocodilite" came to be used to refer to any similarly brain-twisting dilemma where you admit something that is later used against you, while "crocodility" is an equally ancient word for captious or fallacious reasoning

6. THE DICHOTOMY PARADOX

Imagine that you’re about to set off walking down a street. To reach the other end, you’d first have to walk half way there. And to walk half way there, you’d first have to walk a quarter of the way there. And to walk a quarter of the way there, you’d first have to walk an eighth of the way there. And before that a sixteenth of the way there, and then a thirty-second of the way there, a sixty-fourth of the way there, and so on.

Ultimately, in order to perform even the simplest of tasks like walking down a street, you’d have to perform an infinite number of smaller tasks—something that, by definition, is utterly impossible. Not only that, but no matter how small the first part of the journey is said to be, it can always be halved to create another task; the only way in which it cannot be halved would be to consider the first part of the journey to be of absolutely no distance whatsoever, and in order to complete the task of moving no distance whatsoever, you can’t even start your journey in the first place.

7. THE FLETCHER’S PARADOX

Imagine a fletcher (i.e. an arrow-maker) has fired one of his arrows into the air. For the arrow to be considered to be moving, it has to be continually repositioning itself from the place where it is now to any place where it currently isn’t. The Fletcher’s Paradox, however, states that throughout its trajectory the arrow is actually not moving at all. At any given instant of no real duration (in other words, a snapshot in time) during its flight, the arrow cannot move to somewhere it isn’t because there isn’t time for it to do so. And it can’t move to where it is now, because it’s already there. So, for that instant in time, the arrow must be stationary. But because all time is comprised entirely of instants—in every one of which the arrow must also be stationary—then the arrow must in fact be stationary the entire time. Except, of course, it isn’t.

8. GALILEO’S PARADOX OF THE INFINITE

In his final written work, Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638), the legendary Italian polymath Galileo Galilei proposed a mathematical paradox based on the relationships between different sets of numbers. On the one hand, he proposed, there are square numbers—like 1, 4, 9, 16, 25, 36, and so on. On the other, there are those numbers that are not squares—like 2, 3, 5, 6, 7, 8, 10, and so on. Put these two groups together, and surely there have to be more numbers in general than there are just square numbers—or, to put it another way, the total number of square numbers must be less than the total number of square and non-square numbers together. However, because every positive number has to have a corresponding square and every square number has to have a positive number as its square root, there cannot possibly be more of one than the other.

Confused? You’re not the only one. In his discussion of his paradox, Galileo was left with no alternative than to conclude that numerical concepts like more, less, or fewer can only be applied to finite sets of numbers, and as there are an infinite number of square and non-square numbers, these concepts simply cannot be used in this context.

9. THE POTATO PARADOX

Imagine that a farmer has a sack containing 100 lbs of potatoes. The potatoes, he discovers, are comprised of 99% water and 1% solids, so he leaves them in the heat of the sun for a day to let the amount of water in them reduce to 98%. But when he returns to them the day after, he finds his 100 lb sack now weighs just 50 lbs. How can this be true? Well, if 99% of 100 lbs of potatoes is water then the water must weigh 99 lbs. The 1% of solids must ultimately weigh just 1 lb, giving a ratio of solids to liquids of 1:99. But if the potatoes are allowed to dehydrate to 98% water, the solids must now account for 2% of the weight—a ratio of 2:98, or 1:49—even though the solids must still only weigh 1lb. The water, ultimately, must now weigh 49lb, giving a total weight of 50lbs despite just a 1% reduction in water content. Or must it?

Although not a true paradox in the strictest sense, the counterintuitive Potato Paradox is a famous example of what is known as a veridical paradox, in which a basic theory is taken to a logical but apparently absurd conclusion.

10. THE RAVEN PARADOX

Also known as Hempel’s Paradox, for the German logician who proposed it in the mid-1940s, the Raven Paradox begins with the apparently straightforward and entirely true statement that “all ravens are black.” This is matched by a “logically contrapositive” (i.e. negative and contradictory) statement that “everything that is not black is not a raven”—which, despite seeming like a fairly unnecessary point to make, is also true given that we know “all ravens are black.” Hempel argues that whenever we see a black raven, this provides evidence to support the first statement. But by extension, whenever we see anything that is not black, like an apple, this too must be taken as evidence supporting the second statement—after all, an apple is not black, and nor is it a raven.

The paradox here is that Hempel has apparently proved that seeing an apple provides us with evidence, no matter how unrelated it may seem, that ravens are black. It’s the equivalent of saying that you live in New York is evidence that you don’t live in L.A., or that saying you are 30 years old is evidence that you are not 29. Just how much information can one statement actually imply anyway?

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Fake It Until You Make It: 10 Artificial Ruins
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The love of ruins, sometimes called ruinophilia, has for centuries inspired the creation of clever fakes—a host of sham facades and hollowed-out castle shells found on grand English, European, and even American estates. The popularity of constructing artificial ruins was at its peak during the 18th and 19th centuries, but architects occasionally still incorporate them today.

Why build a structure that is already crumbling? Between the 16th and 19th centuries, the popularity of counterfeit ruins was influenced by two factors—a classical education that enforced the ideals of ancient Greece and Rome, and the extended tour of Europe (known as The Grand Tour) that well-to-do young men and women took after completing their education. Travelers might start in London or France and roam as far as the Middle East, but the trip almost always included Italy and a chance to admire Roman ruins. More than a few wealthy travelers returned home longing to duplicate those ruins, either to complement a romantic landscape, to demonstrate wealth, or to provide a pretense of family history for the newly rich.

Here are a few romantic ruins constructed between the 18th and 21st centuries.

1. SHAM CASTLE // BATHAMPTON, ENGLAND

Sham Castle (shown above) is aptly named—it’s only a façade. The "castle," overlooking the English city of Bath, was created in 1762 to improve the view for Ralph Allen, a local entrepreneur and philanthropist as well as to provide jobs for local stonemasons. From a distance it looks like a castle ruin, but it's merely a wall that has two three-story circular turrets and a two-story square tower at either end. The castle is not the only folly (as such purely decorative architecture is often called) that Allen built. He also constructed a sham bridge on Serpentine Lake in what is now Prior Park Landscape Garden—the bridge can't be crossed, but provides a nice focal point for the lake. Today, Sham Castle is part of a private golf course.

2. WIMPOLE FOLLY // CAMBRIDGESHIRE, ENGLAND

Building a structure that looks as if it's crumbling does not preclude having to perform regular maintenance. The four-story Gothic tower known as Wimpole Folly in Wimpole, Cambridgeshire, England, was built 1768-72 for Philip Yorke, first Earl of Hardwicke and owner of the Wimpole Estate. Owned by Britain’s National Trust, the ruin threatened to truly crumble a few years ago, so restoration efforts were needed. The last restoration was so well done it won the 2016 European Union Prize for Cultural Heritage. The Wimpole Estate is now open to the public for walks and hikes.

3. CAPEL MANOR FOLLY // ENFIELD, ENGLAND

Capel Manor at Bulls Cross, Enfield, England has been the site of several grand homes since the estate’s first recorded mention in the 13th century, so visitors might be tempted to believe that the manor house's ruins date back at least a few centuries. But that sense of history is an illusion: The faux 15th-century house was built in 2010 to add visual appeal to the manor gardens, which have been open to the public since the 1920s.

4. ROMAN RUIN // SCHONBRUNN PALACE, VIENNA, AUSTRIA

The Roman Ruin was built as a garden ornament for the 1441-room Schonbrunn Palace in Vienna, one of the most important monuments in Austria. The ruin was once called The Ruins of Carthage, after the ancient North African city defeated by Roman military force. But despite the illusion of antiquity, the ruins were created almost 2000 years after Carthage fell in 146 B.C.E. The ruin’s rectangular pool, framed by an intricate semi-circle arch, was designed in 1778 by the architect Johann Ferdinand Hetzendorf von Hohenberg, who modeled it on the Ancient Roman temple of Vespasian and Titus, which he had seen an engraving of.

5. THE RUINEBERG // POTSDAM, GERMANY

One of the earliest examples of artificial ruins in Germany was the complex of structures known as The Ruinenberg. Frederick the Great, King of Prussia, had a summer palace in Potsdam, near Berlin, that was said to rival Versailles. In 1748 Frederick commissioned a large fountain for the palace complete with artificial ruins. The waterworks part of his plan proved too difficult and was soon abandoned, but not before designer Georg Wenzeslaus von Knobelsdorff constructed the ruins. The complex includes Roman pillars, a round temple, and the wall of a Roman theatre. Since 1927 the site has belonged to the Prussian Gardens and Palaces Foundation, Berlin-Brandenburg.

6. PARC MONCEAU // PARIS, FRANCE

Elegant Parc Monceau is located in the fashionable 8th arrondissement of Paris near the Champs-Elysees and Palais de l’Elysée. In 1778, the Duke of Chartres decided to build a mansion on land previously used for hunting. He loved English architecture and gardens, including the notion of nostalgic ruins, so he hired the architect Louis Carrogis Carmontelle to create an extravagant park complete with a Roman temple, antique statues, a Chinese bridge, a farmhouse, a Dutch windmill, a minaret, a small Egyptian pyramid, and some fake gravestones. The most notable feature of the park is a pond surrounded by Corinthian columns, now known as Colonnade de Carmontelle.

7. HAGLEY PARK CASTLE // WORCESTERSHIRE, ENGLAND

The ruins of the medieval castle at Hagley Park in Worcestershire are definitely fake, but they were built with debris from the real ruin of a neighboring abbey. The folly was commissioned by Sir George Lyttelton in 1747 and designed by Sanderson Miller, an English pioneer of Gothic revival architecture. The castle has a round tower at each corner, but by design only one is complete and decorated inside with a coat of arms. The grounds, which also feature a temple portico inspired by an ancient Greek temple, some urns, and obelisks, are now privately owned and not open to the public.

8. TATA CASTLE RUINS // TATA, HUNGARY

French architect Charles de Moreau (1758-1841) was a scholar of classical Roman architecture known for his ability to counterfeit impressive ruins. Nicholas I, Prince Esterhazy of Hungary, hired him to work on Tata Castle and to create the ruins of a Romanesque church for the palace’s English Garden. Even though the ruin Moreau created was fake, he built it with the stones of a real ruin, the remnants of the early-12th-century Benedictine and later Dominican abbey of Vértesszőlős. A third-century ancient Roman tombstone and relief were placed nearby.

9. BELVEDERE CASTLE // MANHATTAN, NEW YORK

Frederick Law Olmsted and Calvert Vaux designed Central Park in the mid-1800s, and their plan for creating romantic vistas included the construction of a folly known as Belvedere Castle. The Gothic-Romanesque style hybrid, overlooking Central Park’s Great Lawn, was completed in 1869. Although the folly was designed as a hollow shell and meant to be a ruin, it eventually served a practical purpose, housing a weather bureau and exhibit space. The castle also provides a beautiful backdrop for Shakespeare in the Park productions, evoking the royal homes that play prominent roles in the Bard’s works.

10. FOLLY WALL IN BARKING TOWN SQUARE // LONDON

In a borough known for its real historic buildings, the ancient wall found in London’s Barking Town Square might look centuries old. It’s not, and ironically, the wall is part of the square’s renovation efforts. The wall was built by bricklaying students at Barking College using old bricks and crumbling stone items found at salvage yards. Known as the "Secret Garden," named after the children’s book about a walled garden, the wall was designed to screen a nearby supermarket and was unveiled in 2007.

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11 Delicious Facts About Good Burger
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Paramount Pictures

It takes just 14 words—“Welcome to Good Burger, home of the Good Burger, can I take your order?”—to make a ‘90s kid swoon with nostalgia. Good Burger, the beloved Nickelodeon comedy about a couple of daft teens who try to save their fast food joint from corporate greed, was born out of a Kenan Thompson/Kel Mitchell sketch on All That in the mid-'90s. A year later, due to its popularity, it found itself being turned into its own live-action movie, with Brian Robbins at the helm. Today—20 years after its original release—it’s a silly cult hit that’s indelibly a part of Generation Y. Revisit the classic with these facts about Good Burger.

1. KEL MITCHELL AUDITIONED FOR ALL THAT WITH HIS CHARACTER FROM GOOD BURGER.

In an interview with The A.V. Club, Kel Mitchell explained how he came up with Ed. “I did a ‘dude’ voice, and that’s where Ed [from Good Burger] was kind of born,” he said. “I did that there at the audition. They were just cracking up.”

2. ED’S FIRST APPEARANCE WAS IN THE JOSH SERVER SKETCH, “DREAM REMOTE.”

Essentially, Good Burger was born out of a random character decision made during one little sketch. “It was where [Josh] could have a remote control that could control his entire life,” Mitchell told The A.V. Club. “So, he could fast-forward through his sister nagging, he could make pizza come really quickly. I was the pizza guy. I came to the door, and the pizza guy didn’t really have a voice, so I was like, ‘Mleh, here’s your pizza! That was the first time we saw Ed, and so they created Good Burger.”

3. ED’S LOOK WAS INSPIRED BY MILLI VANILLI.

When prepping for Ed’s debut on All That, Kel Mitchell spotted what would become the character’s signature look. “I remember I went to the hair room, and I saw these braids. It was like these early Brandy ’90s Milli Vanilli braids. I put those on, and it came to life,” he told The A.V. Club.

4. THOUSANDS OF POUNDS OF MEAT STUNK UP THE SET.

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For a movie all about burgers, you better believe the production had a ton of them sitting around on set. "At one point, there was over 1750 pounds of meat on the set," Kenan Thompson told The Morning Call. "Some of it was old meat. It was so nasty. Some of the burgers would stay out there for a long time. I felt sorry for the extras who had to eat them with cold, clammy fries. But on screen, those burgers look good."

5. ELMER’S GLUE WAS USED TO KEEP THE FOOD LOOKING FRESH.

In order to keep the food looking good on screen, the production resorted to old, albeit inedible, tricks. "It was so gross, because when I scoop out ice cream in the movie, it was really vegetable shortening with food coloring,” Mitchell told The Morning Call. “When I poured milk on cereal, we used Elmer's Glue so the flakes wouldn't get soggy."

6. KENAN AND KEL CONTRIBUTED TO THE GOOD BURGER SOUNDTRACK.

Good Burger was their baby, so of course Kenan and Kel took the reins on more than just the creation of the characters, according to a 1997 interview with The Morning Call. Specifically, Kel partnered up with Less Than Jake on the hit song, “We’re All Dudes.” Because of this, the soundtrack actually charted at 101 on the Billboard 200.

7. GOOD BURGER WAS LINDA CARDELLINI’S FEATURE FILM DEBUT.

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In an interview with The A.V. Club, the Freaks and Geeks star reminisced about her breakout role in the Nickelodeon movie. “That’s my sister’s favorite role that I’ve ever played! It was so much fun. It was my first film, and it was a fantastic part,” Cardellini said. “I got to play crazy! Nobody knew who I was, and I got the part from the table read.”

8. WRITER DAN SCHNEIDER INTENDED TO GIVE UP ACTING WHEN HE WROTE GOOD BURGER, BUT HE PLAYED MR. BAILY IN THE FILM.

On creating Good Burger, writer/producer/actor Dan Schneider explained to The A.V. Club: “I’ve always wanted to write, and after I was doing All That and Kenan & Kel, I got the opportunity to do another TV show—I was still going on auditions. I realized that if I took that show, I was going to have to give up All That and Kenan & Kel. I really didn’t want to do [that] ... I passed on the acting role, and that was really the turning point, I guess, in 1996, when I was like, ‘You know what? I’m going to put my acting career on the back burner, and I’m going to be a writer-producer.’ Then I wrote the movie Good Burger.” However, if you watch the movie, you’ll notice Schneider starring as Mr. Baily.

9. THE ORIGINAL TRAILER FEATURED A SCENE THAT DIDN’T MAKE THE MOVIE.

For reasons that remain a mystery, a scene where a Good Burger customer orders “a good shake” from Ed (Mitchell), only to receive an actual bodily shaking from the Good Burger employee, didn’t make the final cut. It did, however, feature for a few seconds in the theatrical trailer.

10. KENAN AND KEL REUNITED FOR A GOOD BURGER SKETCH ON THE TONIGHT SHOW.

In 2015, Kenan and Kel reunited for a Good Burger sketch with Jimmy Fallon. This time, however, Fallon played Ed’s co-worker, while Kenan came in as a construction worker as a surprise. "We've been wanting to get back together," Mitchell told E! News. "It was just about the right project ... it felt like home."

11. THE FIRST LINE IN THE FILM IS THE SAME AS THE LAST LINE.

Appropriately, the line is, “Welcome to Good Burger, home of the Good Burger, can I take your order?”—just watch the movie.

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