Tag: numbers

Posted in Science & Nature

Sudoku

Posted on by Jinavie

Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.

The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?

Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.

In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.

The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.

Posted in History & Literature

Baker’s Dozen

Posted on by Jinavie

A dozen is a counting term used to describe 12 of something. But when you have a baker’s dozen of bread, you have 13 pieces of baking, not 12. This may seem like a charitable gift from the baker, but the historical origin is somewhat different.

In the Middle Ages (particulary around the 13th century), baking was not an exact science and loaves of breads were made with varying sizes and weights. This made it easy for bakers to short the customer by giving them smaller loaves than what the customer needed.

To stop this, many countries implemented laws that prevented bakers from shorting the customer, usually by setting a minimum weight for a dozen loaves of bread. However, it is entirely possible for the baker to lose a few loaves of breads to accidental dropping, burning or thieves stealing them. Because the breads may come out smaller, it could not be guaranteed that a dozen loaves would be heavy enough to meet the guidelines – no matter how honest the baker was. To offset this, bakers began adding an extra loaf to ensure that they would not disobey the law (and pay a hefty fine or be seriously punished).

Another theory with less historical evidence is based on the shape of baking trays. Most baking trays are made in a 3:2 ratio and the most efficient way to place loaves of breads on these trays is a 4:5:4 hexagonal arrangement. This arrangement has the advantage of avoiding the corners, where the temperature will heat up then cool down faster, making the results less perfect. Therefore, bakers may have sold a batch of 13 loaves together instead of selling 12 and leaving one out.

Posted in Science & Nature

Interesting Numbers

Posted on by Jinavie

One of the more humorous sides to numbers is mathematicians’ attempts to categorise numbers as “interesting” or “dull”. For example, 1 is interesting because it is the first positive integer. 73 is interesting because it is the 21st prime number and 21 is a multiple of 7 and 3. The number 1729 is a good example of how a number can seem dull but later found to be interesting. When the British mathematician G. H. Hardy visited Indian mathematician Srinivasa Ramanujan, he commented that the number of the taxicab he rode in on was 1729 – a number he found to be rather dull. Ramanujan objected and stated it is very interesting as it is the smallest number expressible as the sum of two cubes in two different ways (1729 = 1³ + 12³ = 9³ + 10³). Such numbers are now referred to as taxicab numbers and 1729 is called the Hardy-Ramanujan number.

A way to discover the smallest most uninteresting number is through the Online Encyclopaedia of Integer Sequences, which documents every integer worth noting as it is in some sort of arithmetic sequence. The smallest integer that does not appear in this encyclopaedia as part of a sequence could be considered as objectively the smallest “uninteresting” number. In 2009, this number was 11630, but has since changed to 12407, then 13794 and now 14228 (22 April 2014).

But paradoxically, the smallest uninteresting number is interesting in itself by being the smallest most uninteresting number. This is known as the interesting number paradox. By this paradox, every natural number is unique and ergo, “interesting”.

Number Line

(Image source: http://www.xkcd.com/899/, and here’s an explanation of some of the numbers on it)

Posted in Science & Nature

Millions And Billions

Posted on by Jinavie

Have you ever stopped and pondered what a million actually is? Sure, you might easily pass it off as the number 1,000,000, or a thousand thousands, but have you really tried to get your head around how big a number that is? For example, you may be able to visualise a hundred people, a thousand people or even tens of thousands of people in your head, but it is very hard to visualise an image of a million people.

Now consider this. When was a million seconds ago? You know a second is very short and a million is a very large number. But it is difficult to put the two together. Make a guess. Last year? Two months ago? Surprisingly, the answer is only a week and a half ago (11.6 days).
Then what about a billion seconds? A billion is a thousand million so you might think it is easy to just add some zeroes, but a billion seconds is 31.7 years ago. Just by changing one syllable, or adding three zeroes, we went from a scale of weeks to years. If we go one step further to a trillion seconds, you leap back in time 31,700 years. You can probably remember what happened a million seconds ago, you might not have even been born a billion seconds ago and our ancestors were still hunter-gatherers roaming Europe a trillion seconds ago. That is how mind-blowing the scale of large numbers can be.

Now let’s look at some other things to really understand how big a million and a billion can be. A million dollars (USD) could buy you a luxury house, a manufacturing line, a 41-acre island in Belize or over 200 years’ worth of coffee (if you drank two cups a day). A million dollars in $1 bills would weigh 1000kg and stack to 30 stories high. A billion dollars – even if you were to convert it into $100 bills – would weigh 10 tonnes, almost as heavy as the truck that would carry it.

The pitter-patter of raindrops on your face feels nice, but a million drops of water weighs 50kg and would break your neck. A billion red helium balloons would have enough lift to carry 14,000 tonnes – enough to lift a hundred small, two-storey houses up into the air. A million grains of rice will feed a person for almost two months, while a billion ants would weigh twice a standard car (3 tonnes total).

Related image

(You should definitely check out Hank Green’s take on “a million seconds”, because everything is better if Hank Green is ranting about it! http://www.youtube.com/watch?v=cJ7A0yTDiqQ)

Posted in Science & Nature

Mathematical Beauty

Posted on by Jinavie

What is the most “beautiful” mathematical equation? For millenia, many mathematical formulas and concepts have been described as beautiful (and some defining beauty, as the golden ratio does). In the mathematical world, the adjective “beautiful” is used in the sense that certain mathematical concepts, despite the fact they are rational and objective, are so pure, simple and elegant that they can only be described as art.

One such formula is Euler’s identity:

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Renowned physicist Richard Feynman described it as “the most remarkable formula in mathematics”. What makes this array of symbols and numbers so beautiful? Firstly, it contains the three basic arithmetic operations exactly once each: addition, multiplication and exponentiation. It also connects five fundamental mathematical constants with nothing other than themselves and the arithmetic operations.

0 is the additive identity, as adding it to another number results in the original number. 1 is the multiplicative identity for the same reason as 0. Pi(π) is one of the most important mathematical constants in the history of mathematics that is ubiquitous in Euclidean geometry and trigonometry. Euler’s number(e) is the base of natural logarithms and is used widely in mathematical and scientific analysis. i(√-1) is the imaginary unit of complex numbers, a field of imaginary numbers that are not “real”, allowing for the calculation of all roots of polynomials. Euler’s identity neatly sums up the relation between these five numbers that are so crucial in the field of mathematics. It is also interesting to note that these five numbers were discovered at different points in history spanning over 3000 years.

Some people describe mathematics as a distinct language in itself. Not only that, but mathematics is considered the universal language as it is both universal and ubiquitous. If that is the case, than Euler’s identity can be considered an extremely pithy literary masterpiece.