142 857 x 1 = 142 857

142 857 x 2 = 285 714

142 857 x 3 = 428 571

142 857 x 4 = 571 428

142 857 x 5 = 714 285

142 857 x 6 = 857 142

If we square 142 857, we have 20408122449, and 20408+122449 = 142 857.

Image Credit: Geek Story

Can you help them solve the problem?

The problem was submitted to Leonard Euler. Euler proved (see simplified proof) that there was no solution to the problem; that is, there was no way that anyone could tour the city without having to skip a bridge or cross a bridge more than once.

The Konigsberg problem, though not that rigorous, gave birth to another branch of mathematics known as Graph Theory. Graph theory is widely used today especially in computer science particularly in networking, data organization, and communication.  It is also used in other sciences such as physics and chemistry to study molecular structures.

What was amazing about the Konigsberg problem was that it came from a practical problem, yet it was so rich that it created a whole universe of its own.

To make it simple, let us use a simple die game for our discussion.  Two dice are rolled. If the sums of the number of dots are 2, 3, 4, 10, 11, and 12, the player wins. If the sums are 5, 6, 7, 8, and 9, the house (the casino) wins.

At first glance, the design seems to favor the player. There are 6 possible sums in favor of him, while there are only 5 possible sums in favor of the house. But looking closely, the design actually favors the house. The secret lies in the mathematics: the number of ways in getting the sums.

The Math Trick

Suppose the dice are colored red and blue. There is only one way to get a sum of 12 (6 + 6), while there are six ways to get a sum of 7: (1 + 6), (2 + 5), (3 + 4), (4 + 3), (5 + 2), (6 + 1). That means we are quite certain to have more 7s than 12s if we roll the two dice a number of times.

The table below shows all the possible sums and how they can be obtained. The first row represents the number of dots on the red die, and the first column represents the number of dots on the blue die.  The  numbers in black texts represent the sums when the number of dots on the two dice are added.

Notice that the yellow squares are wins of players, while the green squares represent wins of the house. There are 36 possible sums — only 12 ways for the players, but 24 ways for the house. You see the difference? In a single roll, the probability that the player winning is 12/36 , and the probability that the casino winning is 24/36.

The design of the game above is equivalent to rolling a painted die — two faces yellow and four faces green (a die has 6 faces), and ignoring the dots. If the top face turns out to be green, then the casino wins; if yellow, then the player wins. Since there are more green faces than yellow faces, it is likely that  a green face will be rolled most of the time.

The principle used in designing the game above is the same principle used in  designing casino games. The designers of the games rely on theoretical probability to ensure the casino winning. That means that even if players win occasionally, they will certainly lose most of the time.

Regular Polygons

Regular polygons are special because they have congruent sides, congruent angles, and have at least one line of symmetry.  Aside from those shown above, we can create more regular polygons, and it is not hard to see that we can create as many as we please.

In three dimensions, we have also regular solids which we call Platonic solids. They are solids whose faces are regular polygons and with the same number of faces meeting at each vertex. Unlike regular polygons, there are only five Platonic solids, and it can be shown mathematically that no other Platonic solids exist.

The Platonic Solids

The Platonic solids were studied extensively in the ancient times, particularly by the Greeks.  In the play Timeaus, Plato associated these solids with the four classical elements —  the octahedron for air, the tetrahedron for fire, the icosahedron for water, and the hexahedron (or cube) for earth. Plato also spoke of the dodecahedron: “There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven”

Four Regular Solids

Despite its antiquity, the beauty of the Platonic solids continue to fascinate many mathematicians and artists of the modern times. M.C. Escher, for example, celebrated them in his woodcut The Four Regular Solids. The woodcut is the intersection of four  Platonic solids with their symmetries aligned.  Escher made it translucent so all solids can bee seen.

Can you see which solid is missing?