You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You’ve got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned.

The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison.

You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned.

You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed.

What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours?

~ solution in two weeks

Ever had a puzzle that looked easy but tortured you incessantly until you found a solution? Would you work on it obsessively for seven years in isolation? Andrew Wiles did just that to prove Fermat’s Last Theorem.

Pierre de Fermat, a famous number theorist of the 17th century, rarely published his work – instead, he would often write comments in the margins of books. In one margin Fermat proposed that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2. However, rather than providing a proof, he only offered this taunting sentence: “I have discovered a truly remarkable proof which this margin is too small to contain.”

The proof for this simple conjecture was not solved for over 350 years and through the centuries became one of math’s greatest puzzles.

Fermat’s Last Theorem has entranced so many mathematicians due to its duality between simplicity and difficulty. It is easy to understand yet almost impossible to prove or disprove. That Fermat claimed to have a proof made it all the more intriguing. Mathematicians made some progress over the centuries in proving the correctness of the theorem for certain values of n – but this was hardly enough to prove that the theorem was correct for all values of n.

Then came along Andrew Wiles. As a child Wiles loved doing math problems. When he was ten he came across Fermat’s Last Theorem which was, at the time, unsolved for 300 years. "It looked so simple, and yet all the great mathematicians in history couldn’t solve it," said Wiles. "I had to solve it."

Wiles quickly became obsessed with solving the problem. Throughout his teenage and college years he worked on it, using his own methods and that of the mathematicians who had worked on it before him. However, when he became a research student he decided to put the problem aside. Wiles realized that current techniques could not solve the problem and that one could spend years without making any progress. Also, a proof of Fermat’s Last Theorem would be completely useless to mathematics – it would not lead to anything useful for mathematicians. Instead, he went on to study elliptical curves at Cambridge.

His study of elliptical curves would prove useful, for in 1986 a new possibility was presented to Wiles. Ken Ribet linked Fermat’s Last Theorem to another unsolved problem, the Taniyama-Shimura conjecture, which happened to be about elliptical curves. If one conjecture was true, both were – thus, if Wiles could prove the Taniyama-Shimura conjecture, he could prove Fermat’s Last Theorem as well.

From that moment on he was determined to solve the riddle. He dropped all other projects he was working on and concentrated on the Taniyama-Shimura conjecture – in secrecy and isolation. “I realized that anything to do with Fermat’s Last Theorem generates too much interest. You can’t really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed.” His wife did not even know that he was working on the problem until he told her during their honeymoon.

Wiles worked on the problem alone for seven years. He devoted all of his time to working on the proof, the only exception being spending time with his family. He had a few breakthroughs but not a complete proof until one day in spring of 1993 in which he had the idea of examining the elliptical curves from the prime five instead of the prime three. Working feverishly (and forgetting to eat lunch), Wiles went to his wife that afternoon saying he had the proof.

Wiles introduced the proof in a series of three lectures which made no mention of Fermat’s Last Theorem, but rather of elliptical curves. However, the audience realized by the end of the third lecture what Wiles was leading them towards. Once Wiles had finished his proof of the Taniyama-Shimura conjecture, he put Fermat’s Last Theorem on the board then concluded saying, “I think I’ll stop there.”

Wiles gained instant fame for having developed a solution to Fermat’s Last Theorem. However, soon Nick Katz discovered that there was an error in a key section of his original proof. This setback proved difficult for Wiles to overcome, and none of the methods he tried could solve the error. He was about to give up when he re-examined his original (though discarded) method and found that there was actually a way to use it to resolve his mistake. “It was so indescribably beautiful,” said Wiles about the moment he solved the problem. “It was so simple and so elegant, and I just stared in disbelief for twenty minutes.” Thus, in 1994 the final proof of Fermat’s Last Theorem was complete, weighing it at 200 pages, more complex than most people can understand.

The full riddle, however, is still not completely solved, for it remains unknown whether Fermat ever really had a brilliant proof to his conjecture. Fermat could not have thought of Wiles’ proof – Wiles says that, "the techniques used in this proof just weren’t around in Fermat’s time." With the many mathematicians who had thought they’d solved it in the past, it is possible that Fermat deluded himself as well – but a simpler solution may still exist. Wiles, however, is content with his difficult proof - “I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream.”

The largest known prime is the largest integer that is currently known to be a prime number.

It was proven by Euclid that there are infinitely many prime numbers; thus, there is always a prime greater than the largest known prime. Many mathematicians and hobbyists enjoy searching for large prime numbers. This may also be profitable as there are several prizes offered by the Electronic Frontier Foundation for record primes.

The use of electronic computers has accelerated the discoveries and found all records since 1951. The record passed one million digits in 1999, earning a $50,000 prize.

As of March 2008, the largest known prime was discovered by the distributed computing project Great Internet Mersenne Prime Search (GIMPS):

You have an 8 by 8 carpet and another 6 by 1 strip of carpet.
                                                 
You are required to cut the 8 by 8 carpet into exactly two pieces, using only one cut, so that these two pieces and the 6 by 1 strip can be used to form a 10 by 7 carpet.

Solution: Before looking at the solution, give this a good good try. It is actually a very interesting problem.

The solution is as follows. This is how we cut the carpet:

Now we translate the two parts 1 unit each in opposite directions, so that the situation looks something like this:

Finally, we translate the part at the right 1 unit upwards, so that we have:

A 6 by 1 area forms in the middle. Now we just fit our 6 by 1 strip of carpet here, and the 10 by 7 carpet gets completed!
 

Next time you’re traipsing around in a wet, sandy, unfamiliar area, you had best be on your guard for the exotic material known as quicksand. If you’re not careful, you could lose your life. Or much more likely, a shoe!

Quicksand is one of the staple environmental hazards in cartoonish low-budget movies, where it has been known to quickly swallow unwitting persons as they wander through jungles and deserts. It is usually represented as a deep pool of sandy goo which blends in with its surroundings, lying in wait to slowly suck in anyone who attempts to traverse it. Barring any nearby vines or branch-wielding comrades, the victim will slowly sink until completely submerged. Unsurprisingly, real quicksand is pretty tame in comparison to Hollywood’s depiction, but traditional quicksand does have an evil cousin that would pose a grave threat to anyone who might stumble upon it… if it exists.

Traditional quicksand is created when water seeps up from an underground source and saturates an area of sand, silt, clay, or any other grainy soil. Normal sand can support extreme amounts of weight because friction between the grains of sand creates a force chain, distributing the load across a large area. But once there is sufficient moisture, the sand and water becomes a suspension where the sand particles are floating within the water. This significantly reduces the friction between the grains of sand, compromising its ability to support weight. Because the water seeps in from the bottom, the top layer of sand is often dry, causing it to appear to be normal sand.

In reality, quicksand is very rarely more than a few feet deep, making it more of a messy nuisance than a life-threatening hazard. Exhaustion is the biggest risk, considering the amount of energy it can take to untangle oneself from the waterlogged soil. When limbs are moved around under the surface, the movement creates a vacuum pressure, greatly increasing the amount of effort necessary to move. The same vacuum effect is what causes certain consistencies of mud to pull the shoes off of your feet.

In the rare instance where quicksand is deep enough that a person could become submerged in it, one would have to try pretty hard to go under. The reason for this is because the human body is a lot more buoyant in quicksand than it is in water, causing a person to float like a cork on the surface.

That’s not to say that there are no dangers to becoming mired in the saturated soil. A person’s buoyancy will be decreased if they are carrying a heavy pack– possibly enough to cause one to sink– so a quick-release backpack is a good idea in areas where quicksand is known to occur. And if a person flails around in quicksand, the sucking effect caused by the moving limbs can cause a person to sink deeper into the goo. There is also the risk of heat exhaustion, animal attacks, rising tide, and other such elemental hazards which may strike as one is working themselves free… not to mention the extremely high risk that sand will become lodged in disagreeable places.

There is another, more sinister flavor of quicksand called dry quicksand which is potentially a lot more dangerous, though there are no confirmed natural occurrences of the phenomenon. Dry quicksand is created when grains of sand form a very loose structure which can barely hold it own weight, like a house of cards. In the lab, it is created by causing air to flow through the sand, but it can theoretically be caused by the gradual buildup of very fine sand after it has been blown into the air. If an object of sufficient weight is placed on the dry quicksand, it will immediately sink, and the delicate structure will rapidly collapse in on itself, burying the object in the process. When this happens, the energy released by the collapse causes a jet of sand particles to shoot high into the air.

A deep, naturally occurring area of dry quicksand would be a formidable hazard, because it would cause anyone who stepped on it to sink and become buried very rapidly. No dry quicksand has ever been officially observed outside of the laboratory, but there are reports of travelers, vehicles, and even whole caravans suddenly vanishing into the sandy earth. These reports have always been viewed as mere folklore, but perhaps there is more to the stories than we realize. Science does not completely dismiss the possibility of naturally occurring dry quicksand; in fact, during the planning of the Apollo moon missions, scientists added large plates to the ends of the Lunar Module legs to help support the craft in case the astronauts found dry quicksand on the moon… but the precaution proved unnecessary, since no such soil was encountered.