There are some mathematical equations in the world that have been tackled by some of the greatest minds but remain unsolved or have taken years to crack. These tasks are a significant challenge for mathematicians around the world who strive to achieve the seemingly impossible. Here is a list of ten such mathematical tasks.

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*1. Archimedes – Stomachion, 250 BC*

*1. Archimedes – Stomachion, 250 BC*

In 1941 Mathematician GH Hardy wrote that “Archimedes remember, and forget the playwright Aeschylus, because languages are dying out, and mathematical ideas live forever.” Indeed, Greek geometer is considered the greatest scientist of antiquity. In 2002 historian of mathematics Revijel Nec comprehended the new way of Archimedes’ work on the puzzle known as “Stomachion”. By studying the ancient text, he realised that the area belongs to the puzzle combinations, branches of mathematics that studies the many ways a problem can be solved. Task “Stomachiona” was to discover how many rotations it would take a shaped puzzle of 14 pieces to agree to make a square. In 2003 mathematicians found a solution: 17,152 rotations.

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**2. Wheat on a chessboard, 1256**

**2. Wheat on a chessboard, 1256**

Sissa chess problem, which is considered the Arab historian Ibn Halikan’s originally, was used for centuries in order to demonstrate geometric progression – one of the oldest chess puzzles. Legend has it that King Shiram offered a reward, which consisted of wheat grains arranged on a chessboard: one grain in the first field, two on the second, three on the third and so on, right up to the last 64th field. However, the king did not realise how much wheat would be required. Turned out 18,446,744,073. 709,551,615 grains of wheat is the answer. This would enough to fill cars that could wrap the Earth 1,000 times.

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**3. Tower of Hanoi, 1883**

**3. Tower of Hanoi, 1883**

The tower of Hanoi was invented by French mathematician Edouard Lucas in 1883 and originally sold as a toy. The aim is that the circles, sorted by the size of a bar (the smallest on top), relocate to another bar in the smallest number of moves. In one move you’re allowed to transfer a single round. It turns out that the minimum number of moves is 2<n> – 1, where n is the number of discs. This means that if you have 64 discs and each move at a speed of one second, finishing the puzzle would take about 585 billion years!

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**4. Rope Around the Earth, 1702**

**4. Rope Around the Earth, 1702**

Imagine you have a rope tightly encircling the equator of a basketball. Now consider how much extra rope you would need for it to be one foot from the surface at all points? Hold that thought and now think about a rope tightly encircling the Earth – making it around 25,000 miles long. Same question: how much extra rope would you need for it to be one foot from the surface at all point? The answer is 2pi (or approximately 6.28) feet for both.If r is the radius of the Earth, and 1 + r is the radius in feet of the enlarged circle, we can compare the rope circumference before (2pir) and after (2pi(1 + r)).

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**5. Prince Rupert’s Problem, 1816**

**5. Prince Rupert’s Problem, 1816**

During the 1600s Prince Rupert of the Rhine came up with a famous, puzzling geometrical question: what is the largest wooden cube that can pass through another cube with one-inch sides? A theory was perhaps a hole can in fact be made in one of two equal cubes that’s sufficiently large for the other cube to slide through – without the cube with the hole falling apart. Mathematicians of today now know though that a cube with a side length of 1.060660 inches (or smaller) can pass through a cube with one-inch sides. This solution was originally discovered by mathematician Pieter Nieuwland and published in 1816.

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**6. Fifteen Puzzle, 1874**

**6. Fifteen Puzzle, 1874**

Today the fifteen puzzle is pretty common and can be purchased in various different forms. In the 19th century however, the fifteen puzzle caused a massive stir. The original game was developed in 1874 by New York postmaster Noyes Palmer Chapman. At the start, the squares show the numbers 1 through 15 in sequence and then a gap. In 1914 a version of the game in Sam Loyd’s Cyclopedia had the 14 and 15 reversed. The goal was to get them them back in the right order. Loyd offered a $1,000 reward for the solution, butit’s impossible to solve the puzzle from this starting position.

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**7. Thirty-Six Officers Problem, 1779**

**7. Thirty-Six Officers Problem, 1779**

Picture this: six army regiments, each consisting of six officers of different ranks. In 1779, Leonhard Euler (he of the Königsberg bridges) asked if it was possible to arrange these 36 o fficers in a 6 × 6 square so that no row or column duplicates a rank or regiment. Euler claimed there was no solution. This puzzle led to many mathematicians attempting to solve the riddle via significant work in combinatorics. Euler also conjectured that this kind of problem could have no solution for an n × n array if n = 4k + 2, where k is a positive integer. This wasn’t settled until 1959, when mathematicians found a solution for a 22 × 22 array.

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**8.Rubik’s Cube, 1974**

**8.Rubik’s Cube, 1974**

Everyone knows of the colourful legend of the Rubik’s Cube. Most of us have probably had a go on one ourselves, some of us may be able to solve it. Invented by Hungarian sculptor and professor of architecture Ernö Rubik in 1974, the colourful puzzle had sold ten million cubes by 1982, which was more than the population of the country. It’s estimated that over 350 million have now been sold worldwide. As we all know the aim of the puzzle is to get the same colour on the face of each side by a number of rotations. In total there are 43,252,003,274,489,856,000 different arrangements of the small cubes. If you had a cube for every one of these ‘legal’ positions, then you could cover the surface of the Earth (including the oceans) about 250 times.

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**9. Barber Paradox, 1901**

**9. Barber Paradox, 1901**

In 1901, the British philosopher and mathematician Bertrand Russell uncovered a possible paradox that necessitated a modification to set theory. One version of Russell’s Paradox involves a town with one male barber who, every day, shaves every man who doesn’t shave himself, and no one else. Does the barber shave himself? The scenario seems to demand that the barber shave himself if and only if he doesn’t shave himself! Russell realised he had to alter set theory so as to avoid such confusion. One way to refute the Barber Paradox might be to simply say that such a barber does not exist. Nevertheless, mathematicians Kurt Gödel and Alan Turing found Russell’s work useful when studying various branches of mathematics and computation.

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