Comment on 10 Mind-Bending Paradoxes That Will Leave You Stumped by 10 Mind-Bending Paradoxes That Will Leave You Stumped – Listverse | C-Suite Mentor©/must visit

Paradoxes can be found everywhere, from ecology to geometry and from logic to chemistry. Even the machine you’re using to read this list has paradoxes of its own. Here are 10 explanations of some of the lesser-known (yet still fascinating) paradoxes of the world. Some concepts are so counterintuitive that we just can’t wrap our minds around them.

Imagine that you are holding a ball. Now picture tearing this ball apart into bits—tear it to pieces, giving the pieces any shape you like. After that, put the pieces back together to form two balls instead of one. How big are these balls in comparison to the one you started with?

Set theoretic geometry would conclude that the matter of the original ball can be separated into two balls of the exact same size and shape as the original ball. Additionally, given two balls of different volume, either ball may be reformed to match the other. This gives way to the cheeky conclusion that a pea may be divided and reshaped into a ball the size of the Sun.

The trick in this paradox is the caveat that you can tear the ball into pieces of any shape. In practice, you cannot really do this—you are limited by the material’s structure and ultimately by the size of atoms. To be able to truly tear the ball however you like, the ball would have to contain an infinite number of accessible zero-dimensional points. The ball would be infinitely dense with these points, and once you separate them, the shapes could be so complex that each would have no defined volume. You could rearrange these shapes, each containing infinite points, into a ball of any size. The new ball would still contain infinite points, and both balls would be equally—infinitely—dense.

Though this idea does not work when you try it on physical balls, it does when you work with mathematical spheres, which are infinitely divisible sets of numbers in three dimensions. The resolution of the paradox, called the Banach-Tarksi theorem, is therefore important to mathematical set theory.

Whales are obviously much larger than we are. This means that they also have far more cells in their bodies. Every cell in the body has the potential to become cancerous. Therefore whales have a higher chance of contracting cancer than we do, right?

Wrong. Peto’s paradox, named after Oxford professor Richard Peto, states that the expected correlation between animal size and cancer prevalence is nonexistent. Humans and beluga whales share a relatively similar chance of getting cancer, while certain breeds of tiny mice have a much higher chance.

Some biologists believe that the lack of correlation in Peto’s paradox comes from tumor-suppressing mechanisms in larger animals. These suppressors work to prevent cell mutation during division.

For something to physically exist, it must be present for a duration of time. Just as an object cannot lack length, width, or depth, it needs duration—an “instantaneous” object, one that doesn’t last for any amount of time, does not exist at all.

According to universal nihilism, the past and future occupy no time within the present. Furthermore, it is impossible to quantify the duration of what we call the present. Any amount of time that you assign to the present can be temporally divided into parts of past, present, and future. If the present is one second long, then that second can be divided into three parts. The first part is then the past, the second part is the present, and the third is the future. The third of a second that is now considered the present can be further divided into three more parts. This division can occur indefinitely.

Therefore, the present can never truly exist as it never occupies a set duration of time. Universal nihilism uses this argument to claim that nothing ever exists.

People have trouble solving problems that require high-level reasoning. On the other hand, basic motor and sensory functions such as walking are no trouble at all. In computers, however, the roles are reversed. It is very easy for computers to process logical problems, such as devising chess strategies, but it takes a lot more work to program a computer to walk or accurately interpret speech. This difference between natural and artificial intelligence is known as Moravec’s Paradox.

Hans Moravec, a research scientist at the Carnegie Mellon University Robotics Institute, explains this observation through the idea of reverse engineering our own brains. Reverse engineering is most difficult for tasks that humans do unconsciously, such as motor functions. Because abstract thought has been a part of human behavior for less than 100,000 years, our ability to solve abstract problems is a conscious one. Therefore, it is much easier for us to create technology that emulates such behavior. On the flip side, actions such as speaking and moving are not ones that we need to actively consider, so it is harder to put these functions into agents of artificial intelligence.

What’s the chance that a random number starts with the digit “1?? Or with the digit “3? or “7?? If you know a little about probability, you would assume that the probability in each case would be one in nine, or about 11 percent.

And yet, if you look at real-world figures, “9? shows up far less than 11 percent of the time. Fewer numbers than expected also start with “8,” while a whopping 30 percent of numbers start with the digit “1.” This paradoxical pattern shows up in all sorts of real measurements, from populations to stock prices to the lengths of rivers.

Physicist Frank Benford first noted this phenomenon in 1938. He found that the frequency of a number appearing as the leading digit drops as the number increases from one to nine. The number one appears as the leading digit approximately 30.1 percent of the time, the number two appears about 17.6 percent of the time, the number three appears about 12.5 percent of the time, and so on until the ninth digit, which appears a mere 4.6 percent of the time.

To explain this, imagine looking at sequentially numbered raffle tickets. Once we’ve noted tickets one through nine, the chance of any number starting with “1? is 11.1 percent. When we add ticket number 10, the chance of a random number starting with “1? goes up to 18.2 percent. As we add tickets 11 through 19, the chance of a ticket starting with “1? keeps rising, peaking at 58 percent. Then when we add ticket 20 and move onward, the chance of a number starting with “2? rises, and the chance of it starting with “1? slowly falls.

Benford’s Law does not apply to every distribution of numbers. For example, sets of numbers that are limited in range, such as human height and weight measurements, do not follow the law. It also does not work with sets that have only one or two orders of magnitude. However, it does apply to many types of data, greatly conflicting with what people would expect. As a result, authorities can use the law to detect fraud. When submitted data does not follow the law, authorities can conclude that someone fabricated the data instead of accurately collecting it.

Genes contain all the information necessary for creating an organism. So it stands to reason that complex organisms would have the most complex genomes—and yet that’s not true at all.

Single-celled amoeba have genomes that are 100 times larger than those of humans. In fact, they have some of the largest genomes that have been observed. Furthermore, species that are very similar to each other can have radically different genomes. This oddity is known as the C-value paradox.

An interesting takeaway from the C-value paradox is that genomes can be larger than necessary. If all of the genomic DNA in humans were in use, the amount of mutations per generation would be incredibly high. The genomes of many complex animals, such as humans and primates, includes DNA that encodes nothing. This huge amount of unused DNA, which varies greatly in quantity from creature to creature, accounts for the lack of correlation that creates the C-value paradox.

Imagine an ant walking the length of a 1-meter (3.3 ft) rubber rope at the rate of 1 centimeter (0.4 in) per second. Imagine that the rope is also being stretched at 1 kilometer (0.62 mi) per second. Will the ant ever make it to the end of the elongating rope?

Logically, it seems impossible for the ant to do so because its rate of motion is far lower than that of its destination. However, the ant will indeed eventually make it to the other side.

Before the ant begins moving, it has 100 percent of the rope left to traverse. After one second, the rope has become considerably longer, but the ant has also moved, decreasing the fraction of rope remaining. Though the distance in front of the ant increases, the small bit of rope that the ant has already covered elongates as well. So though the overall rope lengthens at a steady rate, the distance in front of the ant increases by slightly less each second. The ant, meanwhile, moves forward at a completely steady rate. In this way, with each passing second, the ant chips away at the percentage he still has to cover.

There is one condition needed for this paradox to have a resolution: The ant must be immortal. For the ant to ever make it to the end, it would have to walk for 2.8 x 1043,429 seconds, which exceeds the lifetime of the universe.

Predator-prey models are equations that describe real-world ecological environments. For example, a model may measure how the populations of foxes and rabbits change in a large forest. Suppose the abundance of lettuce increases permanently in the forest. You would expect this to have a good effect on the rabbits that eat lettuce, boosting their population.

The paradox of enrichment states that this may not be the case. The rabbit population rises initially. But the increased density of rabbits in the closed environment leads to an increase in the population of foxes. Rather than finding a new equilibrium, the predators may grow so much in number that they decimate or even wipe out the prey—and thus wipe themselves out as well.

In practice, species may develop means to escape the fate of the paradox, leading to stable populations. For example, the new conditions may induce new defense mechanisms in the prey.

Round up a group of friends, and watch the above video. When it’s over, have everyone say whether the pitch increased or decreased during each of the four pairs of tones. You may be surprised to find that your friends disagree on the answer.

To understand this paradox, you need to know a little about musical notes. A specific note has a specific pitch, which is how high or low it sounds. A note that is one octave above a second note sounds twice as high because its wave has twice the frequency. Each octave interval can be divided into two equal tritone intervals.

In the video, a tritone separates each pair’s sounds. In each pair, one sound is a mixture of identical notes from different octaves—for example, a combination of two “D” notes, one higher than the other. When the sound is played next to a second note one tritone away (for example, a G-sharp between the two D’s), you may validly interpret the second note as either higher or lower than the first.

Another paradoxical application of tritones is an infinite sound that appears to constantly drop in pitch, though it actually cycles continually. This video plays such a sound for 10 hours.

Sitting in front of you are two glasses of water that are identical except for one thing: The water on your left is hotter than the water on your right. Place both of these glasses in the freezer. Which will freeze faster? You’d think the colder glass on the right would, but that might not be the case. Hot water can freeze faster than cold water.

This odd effect is named after a Tanzanian student who observed it in 1986 while freezing milk to make ice cream. But some of history’s greatest thinkers—Aristotle, Francis Bacon, and Rene Descartes—had previously noted this phenomenon without being able to explain it. Aristotle mistakenly attributed it to what he called “antiperistasis,” the idea that a quality intensifies in the environment of its opposite quality.

Several factors contribute to the Mpemba Effect. The hot glass of water may lose a large amount of water from evaporation, leaving less water that needs to be cooled. Warmer water also holds less dissolved gas, which could cause the water to more easily develop convection currents, thereby making it easier for the water to freeze.

Another theory lies in the chemical bonds holding the water molecule together. A molecule of water has two hydrogen atoms bonded to a single oxygen atom. When water heats up, the molecules move apart, and the bonds can relax and give up some of their energy. This lets them cool faster than water that had not been heated to begin with.

Alex Brannan is a student at the University of Wisconsin-Madison and an aspiring fiction writer.

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Comment on 10 Unusual Crime Fighters by 10 Unusual Crime Fighters – Listverse | C-Suite Mentor©/must visit

It takes all different kinds of people to fight crime. There are many different types of law enforcement agencies around the world, and there are even people who dress up in superhero costumes and patrol the streets. However, there are others—ordinary men and women working at normal day jobs—who investigated crimes or mysteries and ultimately brought wrongdoers to justice.

What always amazes adults about young people, especially children, is their natural adaptability to new technology. For example, take eight-year-old Landon Crabtree from Manchester, Tennessee, who managed to bring a serial criminal to justice with just an app.

After Landon’s house was burglarized, he went online and used a tracking app to locate his stolen iPad. The app allowed him to see where his iTunes software was being updated, ultimately showing where his iPad was. Landon’s father alerted the police after his son informed him the missing item had been located at a local motel.

Within an hour of his call, the police tracked down the iPad, along with the rest of the stolen goods and valuables from other houses. The room’s occupant, John Docherty, was arrested on sight. Following his arrest, it was discovered that Docherty was wanted for several other burglaries.

Bill Jakob is the only person on this list who was in law enforcement, though he was unemployed at the time of this story. He was a small-town cop at one time, but he was never a state-certified police officer. He had also worked as a security guard and a wedding minister. According to records, his last known job was as the owner of a trucking company that went bankrupt.

Jakob’s home was 37 kilometers (23 mi) away from the small town of Gerald, Missouri, population 1,172. Like other small towns in America, Gerald was having trouble with methamphetamine. Jakob, calling himself “Sergeant Bill,” went to Gerald to help. He showed the local police a badge and told them he was an undercover agent from the Drug Enforcement Agency. He even produced a phone number they could call to verify his claims. When they called, a woman answered, claiming to work for “the multijurisdictional task force,” and confirmed that Jakob was undercover. Police later came to suspect that the name of the supposed task force had been lifted from the movie Beverly Hills Cop.

Wielding a shotgun, Jakob brought Gerald police along as he started kicking down doors and arresting suspects. When people asked to see a warrant, he told them he didn’t need a warrant because he was a federal agent. He handcuffed people and led them away without reading them their rights. In all, over the span of five months, he arrested about 20 people, and once in custody, most of the suspects confessed.

Jakob’s ruse unraveled when a suspicious reporter began looking into “Sergeant Bill’s” story. He was arrested and pleaded guilty to 23 charges, including impersonating a police officer, and was given a five-year prison sentence. The Chief of Police and two officers were also fired, and the town is now facing lawsuits for wrongful arrests. However, Gerald’s mayor said that at the time, Jakob was “very effective.”

On November 22, 1986, Jose Martinez and his wife, Idalia, were working in their Inwood, New York restaurant, The Dominican Express. At some point during the night, Justo Santos and his two friends began to harass Idalia. Jose kicked them out, and Justo responded by shooting him, leaving Idalia a widow and their nine-year-old daughter, Joselyn, fatherless.

After the murder, Justo fled to the Dominican Republic. The only progress the police made in the investigation was getting Santos to confess on the phone in 1987. He was arrested, but for some reason, he was later released. After that, authorities lost track of Santos, and the murder of Jose looked like it was going to remain unpunished.

That was when Joselyn began investigating the murder that left a giant hole in her life. Her search began in 2005, using paid search engine services to track down her father’s killer. Over the next eight years, Joselyn tracked Justo’s movements and discovered that he was living in Miami, Florida under his own name. Joselyn turned the information over to the NYPD Cold Case Apprehension Squad, and on June 6, 2013, they arrested Justo at his home.

On May 4, 2009, Philip Nisbet died in his home after eating dinner. His wife of eight months, Helen Milner, claimed that Nisbet had committed suicide. He had apparently intentionally induced an allergic reaction by taking a certain medication. Milner proffered a letter that appeared to be Nisbet’s suicide note.

The police seemed to accept her story, and the death of the 47-year-old was ruled a suicide, but Nisbet’s sister, Lee-Anne Cartier, had some reservations. She didn’t think that her brother was a suicidal person, and the signature on the note didn’t match her brother’s. Despite these concerns, police failed to investigate the possibility of foul play, so Cartier took it upon herself. First, she flew from her home in Australia to her brother’s home in Christchurch, New Zealand, to talk to his neighbors and boss. After she took the information she had gleaned to the police, it was revealed that the coroner had noted what appeared to be signs of suffocation on Nisbet’s body. This finally got the police moving on the investigation.

Milner was arrested, convicted of murder, and given an 18-year minimum sentence. The police admitted that they did not handle the case properly and offered to reimburse Cartier for the money she spent on the investigation. She rejected the offer, stating that she instead will be seeking a mysterious “wider compensation.”

On July 11, 2011, eight-year-old Leiby Kletzky went missing on his way home from a day camp in his neighborhood of Boro Park, Brooklyn. The people of Boro Park, a mostly Orthodox Jewish neighborhood, came together to search for the missing boy. One man, a property manager and father of 12 named Yaakov German, was a bit more dedicated than the rest.

German’s son was Kletzky’s teacher, so they started by looking at video surveillance from the school where the day camp was held. The next day, German also persuaded area shopkeepers to let him view their video surveillance. Through the videos, he was able to track the child’s movements. Finally, in footage from outside a car-leasing company, German saw Kletzky talking to a man before getting into his car.

The NYPD tracked the car to the apartment of hardware clerk Levi Aron. The police swarmed Aron’s apartment, and sadly, they found out the boy had been suffocated. Aron confessed to the crime and was given a sentence of 40 years.

In 1983, 88-year-old Gertrude McCabe was brutally murdered in her home. She was beaten, stabbed multiple times, and choked with a bicycle chain. McCabe had no enemies, and since the house was in a state of disarray, the police investigators thought that the murder happened during a burglary. McCabe’s niece, Jane Alexander, wasn’t so sure. Alexander, who was a former intelligence officer in the Navy, thought that the murder might have something to do with a former boyfriend of hers named Tom O’Donnell.

Alexander started seeing O’Donnell at the age of 55 in 1977 after her husband had passed away. Tom was an old family friend, but Jane soon realized no one had any idea who Tom O’Donnell really was. First, O’Donnell convinced Alexander to take out a loan, using her house as collateral. Soon after, he took off with $10,000 of her money. Only after he fled did she look into her personal finances and realize that O’Donnell had left her on the brink of bankruptcy, forcing her to sell her house.

Alexander’s search for O’Donnell eventually led to Las Vegas, where he was arrested for fraud, but no one was able to find enough evidence to link him to McCabe’s murder. After Alexander spent the next 13 years working on the murder case with a police inspector named Jeff Ouimet, two key pieces of evidence finally turned up. The first was a witness, O’Donnell’s nephew, who said that his uncle had mentioned McCabe’s death the day before the police had discovered the body. The second was a police photo Alexander had found that linked O’Donnell to alterations of the crime scene. The investigation was reopened, and O’Donnell was arrested and convicted of first-degree murder with a life sentence. During the trial, prosecutors explained that O’Donnell likely murdered McCabe because he thought that Alexander would receive an inheritance.

On July 29, 2000, the body of 18-year-old Jessica Currin was found on the playing field of a school in Mayfield, Kentucky. Currin had died a horrible death—she had been raped, there was a belt around her neck, and her body was badly burned. The case was given to rookie detective Tom Fortner, who handled it very poorly. A suspect was arraigned and charged with the crime, but he turned out to be innocent. Fortner eventually quit his job with the force and became a security guard, leaving the Currin case to go cold.

It would probably have remained unsolved had it not been for Susan Galbreath. Galbreath was a Mayfield housewife who became obsessed with the murder. She collected information about the case and wrote to many different celebrities, including Julia Roberts and Oprah Winfrey, hoping to garner more media attention. She didn’t have any luck until she wrote to Tom Mangold in April 2004. Galbreath knew Mangold, a veteran journalist with the BBC, from his current events TV show, Panorama.

Amazingly, Mangold agreed to fly to Mayfield to investigate Currin’s case. Mangold and Galbreath interviewed various people attached to the crime, including Jeremy Adams, the man who was erroneously arrested for the crime. During their interviews, one name kept coming up: Quincy Cross, a drug dealer who lived in Tennessee. There was no solid evidence of Cross’s connection to the murder until 2006, when Galbreath received a message on her MySpace page from a woman named Victoria Caldwell admitting that she had helped Cross dump the body. Galbreath connected Caldwell to the police, and she told them the story of how Cross kidnapped, raped, and murdered Currin. Cross was convicted and given a life sentence without parole.

While Galbreath was thanked by some in the aftermath of the trial, she had angered many citizens during her investigation. She eventually had to move out of Mayfair.

In May 2009, Rakesh Singh was at a wedding with his family in Gurgaon, India when the music suddenly stopped. There had been a fatal hit-and-run accident outside the venue. Singh rushed outside to find that his 16-year-old son, Akshay, had been run over by a truck carrying 41 metric tons (45 tons) of mining equipment. The only way Singh knew it was his son was because he recognized his shirt. Otherwise, the teenager’s body was too mangled to be identified.

The heartbroken father demanded justice for his son, but the police did nothing more than file a report, so Singh took matters into his own hands. Taking time off from his job as an industrial consultant, Singh started eating at the roadside dhabas (restaurants) frequented by truck drivers. He even went as far as hitchhiking in order to meet and evaluate the truckers. Singh eventually found his man, Ravinder Kumar, who was promptly arrested by the police.

On June 27, 2005, Judy Weaver was called to the hospital because her 34-year-old son, Ronald Johnson, had fallen into a coma after sustaining blunt force trauma to his head. Witnesses said that Johnson struck his head on the curb when he was trying to do tricks during a bicycle ride on a wet and rainy night.

That didn’t sit well with Weaver. When she saw Ronald on the gurney, she noticed that his pockets had been turned out as if he had been robbed. When she inspected the scene of the accident, she found that the bike was undamaged and there was no mark on the curb where Johnson allegedly struck his head. Weaver kept quiet about these suspicions until she could figure out what was going on. Meanwhile, Johnson succumbed to his injuries nine days later. The detective in charge of the case, Paul “Spike” Hopkins, ruled the death an accident on the basis of witness testimony.

Weaver then received a surprising visit from a 25-year-old man named Jason Gailey, who admitted that he had accidentally punched Johnson. Weaver brought the story to the police, but the case was closed, and they refused to reopen it. With the aid of her daughter, Deborah Moore, Weaver amassed a large file on the witnesses and the death of her son, to little avail.

One night in 2013, a sheriff’s deputy stopped by the restaurant where Weaver was working. Both Weaver and the deputy were having a slow night, so they sat down to talk, and Weaver unloaded for 45 minutes. She talked about all the details she had and slammed the sheriff’s department, especially Spike Hopkins, for their lack of interest in exacting justice for her son. The deputy took notes while he listened and said that he would try to persuade the cold case unit look into the case. That’s when he introduced himself to Weaver—as Lieutenant Paul “Spike” Hopkins.

Hopkins, true to his word, asked the cold case investigators to look into the crime. Using the information gathered by Weaver and Moore, the detectives interviewed the witnesses, and a new story emerged. Apparently, Gailey had been pistol-whipping another man, and Johnson tried to break it up. Gailey struck Johnson with the gun, which is when he fell to the cement and struck his head. Eight years after Johnson’s death, Gailey was arrested and charged with second-degree murder.

In 1988, Todd Matthews became obsessed with a cold case that his wife, Lori, had mentioned to him. In 1968, Lori’s father had come across the body of an apparently murdered girl in Georgetown, Kentucky. The locals nicknamed her “Tent Girl,” and that was the name that was etched into her tombstone. She was a young woman with reddish brown hair and a gap in her front teeth. Other than that, no one knew anything about her.

The case deeply disturbed Matthews, who decided to uncover the true identity of Tent Girl. He scoured library records and police reports, looking for any clue that might help. As the Internet emerged, Matthews saved up to buy a computer specifically to search online message boards and other websites for clues. Matthews soon found he wasn’t alone in his search for a missing person. He came across thousands of people using the Internet for the same purpose.

That’s when the Doe Network was formed. It’s a message board for users to connect reports of missing people with reports of John and Jane Does. Run by volunteers, at first the group was criticized for being overzealous wannabe cops. They implemented screening procedures for the volunteers, and the Doe Network became a strong and useful tool for identifying bodies, succeeding in solving at least 67 cases.

Matthews’s work has led to serving on a national task force, which created the National Missing and Unidentified Persons System (NamUS) in 2007. The NamUS consists of a database of missing people that is cross-referenced with another database of unidentified bodies. The idea is that police departments and medical examiners will update the database and connections will be made.

As for Tent Girl, Matthew finally found what he was looking for on the Internet in 1998. He came across a message from Rosemary Westbrook of Benton, Arkansas, saying that she was looking for her sister, who went missing in 1968. She had reddish-brown hair and a gap in her front teeth. A DNA test confirmed that Weston’s missing sister, Barbara Ann Hackmann, was the same girl found by Lori Matthews’s father. After a 10-year struggle, Todd Matthews finally gave Tent Girl her name back.

Robert Grimminck is a Canadian crime fiction writer. You can follow him on Twitter or Facebook.

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