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| In [[mathematics]], '''Proizvolov's identity''' is an identity concerning sums of differences of [[positive integer]]s. The identity was posed by Vyacheslav Proizvolov as a problem in the 1985 All-Union [[Soviet Student Olympiads]] {{harv|Savchev|Andreescu|2002|p=66}}.
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| To state the identity, take the first 2''N'' positive integers,
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| :1, 2, 3, ..., 2''N'' − 1, 2''N'', | |
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| and partition them into two subsets of ''N'' numbers each. Arrange one subset in increasing order:
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| :<math> A_1 < A_2 < \cdots < A_N. </math>
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| Arrange the other subset in decreasing order:
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| :<math> B_1 > B_2 > \cdots > B_N. </math>
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| Then the sum
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| :<math> |A_1-B_1| + |A_2-B_2| + \cdots + |A_N-B_N| </math>
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| is always equal to ''N''<sup>2</sup>.
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| ==Example==
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| Take for example ''N'' = 3. The set of numbers is then {1, 2, 3, 4, 5, 6}. Select three numbers of this set, say 2, 3 and 5. Then the sequences ''A'' and ''B'' are:
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| :''A''<sub>1</sub> = 2, ''A''<sub>2</sub> = 3, and ''A''<sub>''3''</sub> = 5;
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| :''B''<sub>1</sub> = 6, ''B''<sub>2</sub> = 4, and ''B''<sub>''3''</sub> = 1.
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| The sum is
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| :<math>|A_1-B_1| + |A_2-B_2| + |A_3-B_3| = |2-6| + |3-4| + |5-1| = 4+1+4 = 9, </math>
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| which indeed equals 3<sup>2</sup>.
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| ==References==
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| * {{Citation | last = Savchev | first = Svetoslav | last2 = Andreescu | first2 = Titu | year = 2002 | title = Mathematical miniatures | volume = 43 | series = Anneli Lax New Mathematical Library | publisher = Mathematical Association of America | isbn = 0-88385-645-X}}.
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| ==External links==
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| *[http://www.cut-the-knot.org/Curriculum/Games/ProizvolovGame.shtml Proizvolov's identity] at cut-the-knot.org
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| [[Category:Recreational mathematics]]
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| [[Category:Theorems in number theory]]
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