Above is the entire three-dimensional model starting at zero and finishing at sixty-three. Please note that such is only due to limited visual space since this unique representation of the positive integers continues until infinity. In other words, as the numbers get larger our model increases in both length and width but its maximum height remains constant. For example, the number twenty-nine is represented by the following vertical sequence of objects: an orange cube followed by an empty space and three consecutive pillars (dark green, brown and yellow), respectively. The brown pillar is composed of eight cubes and is the tallest while the orange cube is the shortest. In fact, the brown pillar represents the maximum height of our three-dimensional model as shown above.

So, what is the game about? If you think about these new concepts, it is easy to realize that what we have here is a method of translating or encoding positive integers into physical objects. Given that encoding and decoding are directly associated with the secure transmission of classified information, the objective of our game is to find someone's secret number prior to our opponents. How can we achieve this? Simply by looking at a specific number within the above three-dimensional model and asking questions regarding its colors. The answers must be "yes" or "no" and each player has only three attempts at guessing someone's secret number. If guessed correctly and the number is a prime you get two points otherwise (if the number is composite) you only get one point. The first player who achieves a total of twenty-two or more points is the winner. Let's take the simplest case. Suppose we have two players X and Y where 5 is the secret number of player X and 63 is that of player Y. Assume that player X starts and asks if there is any object colored orange in the physical representation of 63 to which player Y responds negatively. Next, player Y asks X the same question regarding the number 5 and gets "yes" as an answer. Now, player X asks if there is any object colored red to which player Y responds positively. After that player Y takes a wild guess and asks if 5 is the secret number. Finally, player X replies: "You have got it!" and since 5 is a prime number, player Y gets two points. The game restarts with player Y and continues until someone reaches twenty-two or more points. In the case, of more than two players each person attempts to find the secret number of any opponent and is free to direct questions to anyone.

How was such representation generated? In order to answer this question we must learn what is meant by experimental mathematics. Basically, it consists of using computers to help solve mathematical problems. For example, before building the real-life model that you see above, first I translated my mathematical ideas into a computer model which could be easily visualized. Second, I thought about the type of objects to be used in the physical construction of the model itself. Third, I built it and, consequently, I could finally touch the positive integers as if they were physical objects. Please note that the colors present in the adjacent computer model (see any of the three images on your left) and those in the above real-life model are not identical. This is not a problem since the choice of color for each distinct physical object is arbitrary. In conclusion, the shape of our infinite three-dimensional model is fixed with respect to length, width and height but the selection of colors can vary. Take a look (second image on your left) at the number one in the adjacent computer model represented by a single dark-blue cube while the number sixty-three (look at the third image on your left) consists of six consecutive pillars (purple, light-blue, yellow, yellow, yellow and light-blue) vertically displayed in such order. In this computer model, we can clearly see that the purple, light-blue and yellow pillars are composed of two, four and five identical cubes, respectively.

Experimental mathematics can be used for a wide variety of reasons: test conjectures; confirm analytically derived results; explore different approaches for formal proof; replace lengthy hand calculations; graph and model to expose mathematical facts and structures as we have just seen or simply to gain insight and intuition. In addition, I have used it extensively in my independent research to discover new facts, patterns and relationships concerning the prime numbers. For example, first I assumed that every twin prime is either a pair of an ordinary prime and a Gaussian prime or vice-versa. Second, using two different computers, I confirmed my conjecture for the first eighty million twin primes. Third, given such computational evidence I went ahead and formally proved that every twin prime is either a pair of an ordinary prime and a Gaussian prime or vice-versa.

If you have a problem but do not know how to solve it, perhaps experimental mathematics can help. In any case, please do contact me.