In order to decide which team should kick off, a referee at a soccer game usually tosses a coin into the air and checks if it lands heads or tails. At a casino poker table, a croupier rolls a die and checks how many dots are embossed on the top side when it stops rolling. In the game of Lotto, a blast of air spews a bunch of Ping-Pong balls with numbers inscribed on them high into the air. At some point in time the machine swallows one of the balls and its number is recorded. In each of these cases, one may safely assume that the result of the throw was arrived at by pure chance. One can never predict the face of the coin, the dots on the die, or the number on the Ping-Pong ball.

Purists may point out that one side of a die or a coin might be heavier than the other and that this ever so slight weight difference may distort the results. But apart from this minor flaw, these objects generate fairly acceptable strings of random numbers—between 0 and 1 for coins, between 1 and 6 for dice, and between 1 and 45 for lottery Ping-Pong balls.

Random numbers are important not only in games or sports. There are several other areas in which these numbers are indispensable tools of the trade. In cryptography, for example, random numbers (actually randomly chosen prime numbers) are used to encrypt data. In engineering or economics random numbers are used for simulations. Instead of elaborately calculating a city’s transportation flow by using probability theory, traffic situations can be tested with the help of simulations. A computer program could be written where a particular traffic light turns green whenever a randomly selected number is between 16 and 32, a truck approaches from the left, if the random number is odd, and so on. The simulation is run several

thousand times, and an operator notes the observations: Do accidents occur? Are traffic jams caused? Since random numbers are reminiscent of the roulette game, such methods became known as Monte Carlo simulations. Even mathematics, the most exact science of all, can benefit from simulations. The volumes of complicated shapes, for example, can be determined by calling on Monte Carlo methods.

There is a problem, though, when generating random numbers with the methods described above. Techniques that involve throwing coins, dice, Ping-Pong balls, and other objects into the air are extremely inefficient. At best, about one number can be generated per second. To run high-quality simulations, millions and sometimes even billions of random numbers are required. It stands to reason that one would use computers to generate random numbers. After all they are able to produce huge amounts of numbers in just fractions of a second. But there is a snag. One of the great advantages of electronic computers—the ability to mindlessly execute prescribed instructions over and over again—turns out to be a devastating handicap when it comes to generating strings of random numbers. Starting with any number, computers will always calculate the next number, based on the preceding value. This means that in a sequence of computer-generated random numbers patterns will appear, and every number can, in principle, be predicted. The formula used to generate the “random” numbers may be complex and the pattern may be complex, but a pattern there is nevertheless. For good reason, computer-generated random numbers are called pseudorandom numbers. Even though they may pass a battery of tests for randomness, they are not truly random.

The trick that is used in the creation of pseudorandom numbers by computer involves using a random starting value, or seed. Once a seed has been selected, the program proceeds in a deterministic but unfathomable (to the user) way. It may calculate something along the lines of “take the third root of the preceding number, divide the result by 163, and then pick the 7th, 12th, and 20th

digits after the decimal point.” With this three-digit pseudorandom number in hand, the subsequent pseudorandom number can be calculated, and so on. Of course, since this example involves only three digits, no more than 1,000 different pseudorandom numbers can be generated. And whenever the computer encounters a three-digit number that has been used before, the program will proceed from there with an identical string of numbers. Hence cycles are invariably produced. Their onset can be delayed by making the pseudorandom numbers 15, 20, or more digits long, but in the end even the longest pseudorandom number sequence will end up in a cycle.

Whatever the size of the pseudorandom number, it is imperative that the signal which starts the process come from outside the computer. Otherwise the procedure would always start off with the same seed and all sequences generated by this program would be identical. Many things may serve as a starting signal: the time when the computer operator hits the “Enter” button on the keyboard; the operator’s imperceptible, hence random, hand movement when he or she moves the computer mouse; and so forth.

But however carefully thought out the process might be, in the end all computer-generated random number sequences are of the “pseudo” kind. Scientists nevertheless thought that they could obtain satisfactory results and used their random number generators without many questions. In 1992, however, three physicists found to their horror that their simulations produced incorrect predictions, and hence the conclusions derived from their work were erroneous. Things got even worse. In 2003, two German physicists, Heiko Bauke and Stephan Mertens, proved that generators of random binary numbers produced too many zeros and not enough ones, due to the special role played by zero in algebra.

Organizations specializing in random numbers saw an opportunity. They decided to generate not only the starting value but all numbers outside the computer. The resulting strings of random numbers are put at the disposal of interested parties via the World Wide Web. The sources

for these random numbers are natural phenomena such as the thermal crackling of transistors, the decay of radioactive material, the blubbering of lava lamps, or atmospheric background noise—all of which are completely, undeniably, incontestably random. These phenomena can be measured and registered with the help of Geiger counters, thermometers, or megaphones. Hence, genuine random numbers are being generated and not their “pseudo” versions.