You and nine other individuals have been captured by super-intelligent alien overlords. The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative beings. Unfortunately, they’re not sure whether you qualify, so they decide to give you all a test.
Online gaming can be a fun pastime, and it is for millions of people all over the world. But, how many people ever stop to wonder how this system works. For example, how are mathematics and probability are used in online gambling software? Having this information at your fingertips can help you gain a deeper understanding of online gaming and may increase your chances of success.
Online casinos use online online gaming software to calculate a collection of probability applications that are used in game theory. To do so, this software uses different formulas, like the Fundamental Formula of Gambling. This formula calculates the number of things that must be done for an event of probability to appear with a degrees of certainty. For example, the Fundamental Formula of Gambling can be use to figure out the number of coin tosses that would be necessary to get at least one “tails” with a degree of certainty. The answer is 7 tosses.
Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets
Royal Vegas is an online casino that has been in businesses since 2000. Since then, they have developed a stellar reputation of being a trustworthy site with a large selection of casino games (over 600) for users to try out.
A part of this purveyor of award-winning casino entertainment’s success is the use of Microgaming software to keep their site running smoothly. Microgaming is an online gaming industry leader that provides online gaming site with a software infrastructure that is stable and reliable.
They have been in business for over twenty years because of their reputation for providing a quality product. For instance, their large selection of games has some of the best sounds and graphics on the market. This quality and high level of security ensure that the customers of online gaming sites, like Royal Vegas, will enjoy a gaming experience in a fully secure and virus-free gaming environment.
Can you use math to get you and your friends over the bridge before the zombies arrive?
It is night. Four scientists are escaping from zombies. They have one lantern between them. They reach a gorge, spanned by a log bridge. They need to cross the gorge before the zombies arrive, and then destroy the bridge.
As with all such puzzles, there are various artificial constraints:
– The scientists have 17 minutes before the zombies arrive, so they must cross the gorge in less than 17 minutes.
– Only two people can be on the bridge at the same time or it will not hold them.
– They need the lantern to cross – they can’t cross in the dark (It’s a long bridge and an old-fashioned lantern. You can barely see your feet; you can’t see very far ahead or behind on the bridge). But it’s OK to be left alone in the dark on one side or the other.
It gets more complicated:
– Person A takes 1 minute to cross
– Person B takes 2 minutes to cross
– Person C takes 5 minutes to cross
– Person D takes 10 minutes to cross
Your job is to get everyone across the bridge, never more than two on the bridge at a time, always a lantern on the bridge, in under 17 minutes so they outwit the zombies.
Most people will probably remember the times tables from primary school quizzes. There might be patterns in some of them (the simple doubling of the 2 times table) but others you just learnt by rote. And it was never quite clear just why it was necessary to know what 7 x 9 is off the top of your head.
Well, have no fear, there will be no number quizzes here.
Instead, I want to show you a way to build numbers that gives them some structure, and how multiplication uses that structure.
Multiplication simply gives you the area of a rectangle, if you know the lengths of the sides. Pick any square in the grid, (for example, let’s pick the 7th entry in the 5th row) and colour a rectangle from that square to the top left corner.
This rectangle has length 7 and height 5, and the area (the number of green squares) is found in the blue circle in the bottom right corner! This is true no matter which pair of numbers in the grid you pick.
Now let’s take this rectangle and flip it around the main diagonal (the red dotted line).
The length and height of the rectangle have swapped, but the area hasn’t changed. So from this we can see that 5 × 7 is the same as 7 × 5. This holds true for any pair of numbers — in mathematics we say that multiplication is commutative.
But this fact means that there is a symmetry in the multiplication table. The numbers above the diagonal line are like a mirror image of the numbers below the line.
So if your aim is to memorise the table, you really only need to memorise about half of it.
The building blocks of numbers
To go further with multiplication we first need to do some dividing. Remember that dividing a number just means breaking it into pieces of equal size.
12 ÷ 3 = 4
This means 12 can be broken into 3 pieces, each of size 4.
Since 3 and 4 are both whole numbers, they are called factors of 12, and 12 is said to be divisible by 3 and by 4. If a number is only divisible by itself and 1, it is called a prime number.
But there’s more than one way to write 12 as a product of two numbers:
12 × 1
6 × 2
4 × 3
3 × 4
2 × 6
1 × 12
In fact, we can see this if we look at the multiplication table.
The number of coloured squares in this picture tells you there are six ways you can make a rectangle of area 12 with whole number side lengths. So it’s also the number of ways you can write 12 as a product of two numbers.
Incidentally, you might have noticed that the coloured squares seem to form a smooth curve — they do! The curve joining the squares is known as a hyperbola, given by the equation a × b = 12, where ‘a’ and ‘b’ are not necessarily whole numbers.
Let’s look again at the list of products above that are equal to 12. Every number listed there is a factor of 12. What if we look at factors of factors? Any factor that is not prime (except for 1) can be split into further factors, for example
12 = 6 × 2 = (2 × 3) × 2
12 = 4 × 3 = (2 × 2) × 3
No matter how we do it, when we split the factors until we’re left only with primes, we always end up with two 2’s and one 3.
2 × 2 × 3
is called the prime decomposition of 12 and is unique to that number. There is only one way to write a number as a product of primes, and each product of primes gives a different number. In mathematics this is known as the Fundamental Theorem of Arithmetic.
The prime decomposition tells us important things about a number, in a very condensed way.
For example, from the prime decomposition 12 = 2 × 2 × 3, we can see immediately that 12 is divisible by 2 and 3, and not by any other prime (such as 5 or 7). We can also see that it’s divisible by the product of any choice of two 2’s and one 3 that you want to pick.
Furthermore, any multiple of 12 will also be divisible by the same numbers. Consider 11 x 12 = 132. This result is also divisible by 1, 2, 3, 4, 6 and 12, just like 12. Multiplying each of these with the factor of 11, we find that 132 is also divisible by 11, 22, 33, 44, 66 and 132.
It’s also easy to see if a number is the square of another number: In that case there must be an even number of each prime factor. For example, 36 = 2 × 2 × 3 × 3, so it’s the square of 2 × 3 = 6.
The prime decomposition can also make multiplication easier. If you don’t know the answer to 11 × 12, then knowing the prime decomposition of 12 means you can work through the multiplication step by step.
11 x 12
= 11 x 2 × 2 × 3
= ((11 x 2) × 2) × 3
= (22 × 2) × 3
= 44 × 3
If the primes of the decomposition are small enough (say 2, 3 or 5), multiplication is nice and easy, if a bit paper-consuming. Thus multiplying by 4 (= 2 x 2), 6 (= 2 x 3), 8 (= 2 x 2 x 2), or 9 (= 3 x 3) doesn’t need to be a daunting task!
For example, if you can’t remember the 9 times table, it doesn’t matter as long as you can multiply by 3 twice. (However this method doesn’t help with multiplying by larger primes, here new methods are required – if you haven’t seen the trick for the 11 times tables watch this video).
So the ability to break numbers into their prime factors can make complicated multiplications much simpler, and it’s even more useful for bigger numbers.
For example, the prime decomposition of 756 is 2 x 2 x 3 x 3 x 3 x 7, so multiplying by 756 simply means multiplying by each of these relatively small primes. (Of course, finding the prime decomposition of a large number is usually very difficult, so it’s only useful if you already know what the decomposition is.)
But more than this, prime decompositions give fundamental information about numbers. This information is widely useful in mathematics and other fields such as cryptography and internet security. It also leads to some surprising patterns – to see this, try colouring all multiples of 12 in the times table and see what happens. I’ll leave that for homework.
Physicist Werner Heisenberg said, “When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first.” As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks. Natalya St. Clair illustrates how Van Gogh captured this deep mystery of movement, fluid and light in his work.
Lesson by Natalya St. Clair, animation by Avi Ofer.