Pi (π) a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159, but since it is an irrational number (cannot be expressed as a ratio), the decimal places go on and on with no repeating segments. The history of pi extends back to almost 5000 years ago, as it plays such a crucial role in geometry, such as finding the area of a circle (A = π ²). It is not an understatement to say that pi is among the top five most important numbers discovered in history (0, 1, i and e being the others).
The interesting thing about pi is that it is an irrational number. As mentioned above, this means that pi has an infinite number of non-repeating decimal places, with numbers appearing in random sequence. For example, pi to a 30 decimal places is 3.141592653589793238462643383279… Because of this feature, pi contains all possible sequences and combinations of numbers at a certain point. The corollary to this fact is, if pi is converted into binary code (a number system of only 0 and 1, used by computers to encode information), somewhere in that infinite string of digits is every combination of digits, letters and symbols imaginable. The name of every person you will ever love. The date, time and manner of your death. Answers to all the great questions of the universe. All of this is encoded in one letter: π.
That, is the power of infinity.
The victory condition for this card game, named after an ancient Greek city, is quite simple: discover the pre-determined law via induction.
This game needs at least four people, with one person acting the position of God. God decides on a certain law (in the form of a single statement) and writes it down on a paper, thus creating the way of the universe.
Next, the deck of cards is split evenly between the other players, then one person places a card in the centre. After “the world begins to exist”, God looks at the card and says “This card qualifies” or “This card fails”. The next player also places a card in the centre and the God judges whether it fits the way of the universe.
Players carefully study which cards qualify or fail to try discover the way of the universe. If someone thinks they figured the law out, he or she proclaims themself as the prophet, who begins to take over the role of God to announce whether the cards qualify or not. If at any point the prophet is wrong, he is dismissed. If the prophet correctly judges ten cards in a row, he states his hypothesised law and compares it to the piece of paper. The prophet wins if the two laws coincide, but is dismissed if it is not. When all 52 cards are played without a successful prophet, God becomes victorious.
However, as this is a game, the way of the universe cannot be too complex. To make it fun, the God player must devise a law that is simple yet difficult to discover. For example, the law “Alternate a card higher than 9 and a card lower than 9” is tricky as players tend to focus on picture cards or the colours of the cards. Also, laws such as “Only red cards qualify, except the tenth and thirtieth cards are disqualified” and “Accept all cards that are not the 7 of Hearts” are illegal, as they are too detailed. A God who comes up with such ways of the universe that cannot be found using logic and the scientific method loses his right to play the game. Ergo, the God must seek simplicity that is not easily conceived.
So what is the most successful strategy for this game? Even if there is the risk of being dismissed, proclaim yourself as the prophet as soon as possible for the best chance of winning.
Find the rule for the following sequence of numbers:
Continue reading “Ant Sequence”
1 1 2 3 5 8 13 21 34 55…
A keen observer would note that each number in the above sequence is the sum of the two numbers before it. These are known as Fibonacci numbers and are among the most famous number sequences in mathematics.
It is famous because of some unique properties. For example, every third number is even, every xth number is the multiple of Fx (e.g. 4th number = 3, 8th number = 21…) and the list goes on. It is also known to approximate golden spirals, a mathematical function that is closely related with yet another famous number: the golden ratio.
However, a more interesting (and more relatable) fact about these numbers is that they appear repeatedly in nature. It has been noted for many centuries that plants tend to follow the Fibonacci sequence in various ways. This includes the number of branches of trees that grow per year, the number of petals on a flower (almost all flowers have a Fibonacci number of petals) and most interesting of all: the arrangement of florets on the face of a sunflower. If one carefully scrutinises the face of a sunflower (also applies to pine cones), they will note that the florets (tiny pieces on the face) are arranged in what appears to be spirals. They are actually arranged on a stack of spirals, both clockwise and anti-clockwise. The number of spirals for both directions are always two Fibonacci numbers next to each other (e.g. 34 and 55).
This is because natural selection pushes the plants to arrange their florets, petals and tree branches in the most efficient manner possible, which is provided by the Fibonacci sequence.