The paradox is illustrated by two games played with coins weighted on one side so that they will not fall evenly by chance to heads or tails.

In game A, a player tosses a single loaded coin and bets on each throw. The probability of winning is less than half. In game B, a player tosses one of two loaded coins with a simple rule added. He plays Coin 1 if his money is a multiple of a particular whole number, like three.

If his money cannot be divided by the number three, he plays the Coin 2. In this setup, the second will be played more often than the first.

Both are loaded, one to lose badly and one to win slightly, with the upshot being that anyone playing this game will eventually lose all his money.

“Sure enough,” Dr. Abbott said, when a person plays either game 100 times, all money taken to the gambling table is lost. But when the games are alternated — playing A twice and B twice for 100 times — money is not lost.

It accumulates into big winnings. Even more surprising, he said, when game A and B are played randomly, with no order in the alternating sequence, winnings also go up and up.

This is Parrando’s paradox.

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Euler’s formula

Euler’s formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then

v − e + f = 2.

Seven Bridges of Königsberg

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology.

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot. Euler proved that the problem has no solution. There could be no non-retracing the bridges. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

Euler’s identity

In analytical mathematics, Euler’s identity (also known as Euler’s equation), named for the Swiss mathematician Leonhard Euler, is the equality

where
e is Euler’s number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = −1, and
π is pi, the ratio of the circumference of a circle to its diameter.

Anscombe’s quartet comprises four datasets that have nearly identical simple statistical properties, yet appear very different when graphed. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data before analysing it and the effect of outliers on statistical properties.

The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two variables correlated and following the assumption of normality. The second graph (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third graph (bottom left), the distribution is linear, but with a different regression line, which is offset by the one outlier which exerts enough influence to alter the regression line and lower the correlation coefficient from 1 to 0.816. Finally, the fourth graph (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

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In this article I shall discuss several game programming techniques, all revolving around a central theme: the sine and cosine functions. This article will explain sine, cosine, vectors, atan2, and some useful special effects such as how to make homing missiles and how bitmap rotation works. I shall start with the very basics, but later on I’ll cover some more advanced programming techniques.
Let’s start with something that can sometimes be difficult to understand for beginners, because it is highly abstract – the vector. A vector can be visualized in different ways.

First of all, you can imagine it as an arrow to a point in space. In the case of two-dimensional space, you need two values to define the vector – one for the x-coordinate and one for the y-coordinate. In the case of three-dimensional space, you will need a third value for the z-coordinate. This article will mostly deal with two-dimensional space though. Three-dimensional space is more complicated, and I’m no expert in that.

In the figure above I have drawn a vector with an x-coordinate of 3 and a y-coordinate of 2. But these two values are not the end of the story. For example, if you draw this vector out on paper, you can measure the length of the vector as 3.6 and the angle between the vector and the x-axis as 34 degrees.

If you think about this further, you can see that you don’t even need the (x,y) coordinates of the vector if you already know its length and the angle it makes with the x-axis. It is perfectly possible to define a vector fully just by its length and angle.

If you use the x- and y-coordinates, you are using Cartesian coordinates. If you use the angle and length of the vector, you are using polar coordinates.

Let’s have an example. Suppose you are writing a top-down racing game (something like Micro Machines). You will need a way to store the velocity (speed and direction) of a racing car. And how do we do that? With a vector. This velocity vector is in fact the change in the racing car’s position from one frame to the next (see figure below). The question is, should we use Cartesian coordinates or polar coordinates for this vector?

Well, storing only the Cartesian coordinates has the advantage that it is very easy to calculate the new position of the racing car at each step. Suppose you store the (x,y) coordinates of the velocity vector in the variables vel_x and vel_y, and the position of the racing car in the variables pos_x and pos_y. All you need to do in the game loop is:

pos_x += vel_x;
pos_y += vel_y;

What could be simpler?

On the other hand, storing the length and angle of the velocity vector has its advantages, in that it makes it easier to implement the racing car controls. Think about it – if the player presses LEFT, you want the racing car to turn left. Supposing you store the angle in the integer car_angle, you could use the following code:

if (key[KEY_LEFT])
{
car_angle -= 1; // turn one degree to the left
}
if (key[KEY_RIGHT])
{
car_angle += 1; // turn one degree to the right
}

And how would you do that if you only stored x and y? You would need to change both of them, but how? That is a lot more difficult! Furthermore, if the player presses UP you want the racing car to go faster. You can achieve this by simply increasing the length of the vector. If you store x and y, you have to change both of them a bit, which is again more complicated.

If a card shows an even number on one face, then its opposite face is blue.

When the Weaver brought out a square piece of beautiful cloth, daintily embroidered with lions and castles, as depicted in the illustration, the pilgrims disputed among themselves as to the meaning of these ornaments. The Knight, however, who was skilled in heraldry, explained that they were probably derived from the lions and castles borne in the arms of Ferdinand III., the King of Castile and Leon, whose daughter was the first wife of our Edward I. In this he was undoubtedly correct. The puzzle that the Weaver proposed was this. “Let us, for the nonce, see,” saith he, “if there be any of the company that can show how this piece of cloth may be cut into four several pieces, each of the same size and shape, and each piece bearing a lion and a castle.” It is not recorded that anybody mastered this puzzle, though it is quite possible of solution in a satisfactory manner. No cut may pass through any part of a lion or a castle.

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