What is the Middle Square Method ?

The Middle Square Method is a simple pseudo-random number generator developed by John von Neumann. It's one of the earliest techniques for generating random numbers. The method involves squaring a number and then taking the middle part of the result as the next number in the sequence.

How It Works

Start with an initial seed: This is the initial number you begin with.

Square the seed: Compute the square of the seed.

Extract the middle digits: Select the middle part of the squared number as the next seed.

Repeat the process: Use the new seed to generate the next number.

In mathematics and computer science, the middle-square method is a method of generating pseudorandom numbers. In practice it is a highly flawed method for many practical purposes, since its period is usually very short and it has some severe weaknesses; repeated enough times, the middle-square method will either begin repeatedly generating the same number or cycle to a previous number in the sequence and loop indefinitely.

The method was invented by John von Neumann, and was described by him at a conference in 1949.[1]

In the 1949 talk, Von Neumann quipped that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." What he meant, he elaborated, was that there were no true "random numbers", just means to produce them, and "a strict arithmetic procedure", like the middle-square method, "is not such a method". Nevertheless, he found these methods hundreds of times faster than reading "truly" random numbers off punch cards, which had practical importance for his ENIAC work. He found the "destruction" of middle-square sequences to be a factor in their favor, because it could be easily detected: "one always fears the appearance of undetected short cycles".[1] Nicholas Metropolis reported sequences of 750,000 digits before "destruction" by means of using 38-bit numbers with the "middle-square" method.[2]

The book The Broken Dice by Ivar Ekeland gives an extended account of how the method was invented by a Franciscan friar known only as Brother Edvin sometime between 1240 and 1250.[3] Supposedly, the manuscript is now lost, but Jorge Luis Borges sent Ekeland a copy that he made at the Vatican Library.

Modifying the middle-square algorithm with a Weyl sequence improves period and randomness.[4][5]

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