Editing Middle-square method (section)

===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>