Middle-square method: Difference between revisions

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{{Short description|Pseudorandom number generator}}

[[Image:Middle-square method.svg|right|250px|thumb|One iteration of the middle-square method, showing a 6-digit seed, which is then squared, and the resulting value has its middle 6 digits as the output value (and also as the next seed for the sequence).]]

[[Image:Middle_square_method_2_digits.svg|right|250px|thumb|[[Directed graph]] of all 100 2-digit pseudorandom numbers obtained using the middle-square method with ''n'' = 2.]]

In [[mathematics]] and [[computer science]], the '''middle-square method''' is a method of generating [[pseudorandom number]]s. In practice it is not a goodhighly flawed method for many practical purposes, since its [[Pseudorandom number generator#Periodicity|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.

== History ==

===In mathematics===

The method was invented by [[John von Neumann]], and was described by him at a conference in 1949.<ref name="vonneumann">The 1949 papers were not reprinted until 1951. John von Neumann, “Various techniques used in connection with random digits”, in A.&nbsp;S. Householder, G.&nbsp;E. Forsythe, and H.&nbsp;H. Germond, eds., ''Monte Carlo Method, National Bureau of Standards Applied Mathematics Series'', vol.&nbsp;12 (Washington, D.C.: U.S. Government Printing Office, 1951): pp.&nbsp;36–38.</ref>

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".<ref name="vonneumann"/> [[Nicholas Metropolis]] reported sequences of 750,000 digits before "destruction" by means of using 38-bit numbers with the "middle-square" method.<ref>Donald E. Knuth, ''The art of computer programming, Vol.&nbsp;2, Seminumerical algorithms'', 2nd edn. (Reading, Mass.: Addison-Wesley, 1981), ch.&nbsp;3, section&nbsp;3.1.</ref>

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.<ref name="Ekeland1996">{{cite book |author=Ivar Ekeland |title=The Broken Dice, and Other Mathematical Tales of Chance |date=15 June 1996 |publisher=University of Chicago Press |isbn=978-0-226-19992-4}}</ref> 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.<ref>{{cite book | title = Random Numbers and Computers | last = Kneusel | first = Ron | publisher = Springer | year = 2018 | edition = 1 | pages = 13–14 }}</ref> <ref>{{cite arXiv | last=Widynski | first=Bernard | eprint=1704.00358 | title=Middle-Square Weyl Sequence RNG | date=April 2017| class=cs.CR }}</ref>

== The method ==

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=== Example implementation ===

Here, the algorithm is rendered in [[Python 3|Python 3.12]].

<syntaxhighlight lang="python">

seed_number = int(input("Please enter a four-digit number:\n[####] "))

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f" with {number}.")

</syntaxhighlight>

==References==

<references/>

==See also==

* [[Linear congruential generator]]

* [[Blum Blum Shub]]

* [[Hash function#Mid-squares | middle-square hash function ]]

==References==

<references/>

{{DEFAULTSORT:Middle-Square Method}}

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[[Category:Articles with example C code]]

[[Category:Articles with example Python (programming language) code]]

[[Category:John von Neumann]]