Proizvolov's identity

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The following defines Proizvolov's identity.

Take the first 2N natural numbers:

1, 2, 3, ..., 2N − 1, 2N.

Choose any subset of N numbers and arrange them in an increasing sequence:

A_1 < A_2 < \cdots < A_N.

Arrange the remaining numbers in a decreasing sequence:

B_1 > B_2 > \cdots > B_N.

Then the sum

|A_1-B_1| + |A_2-B_2| + \cdots + |A_N-B_N|

is always equal to N2.

[edit] Example

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:

A1 = 2, A2 = 3, and A3 = 5;
B1 = 6, B2 = 4, and B3 = 1.

The sum is

| A1B1 | + | A2B2 | + | A3B3 | = | 2 − 6 | + | 3 − 4 | + | 5 − 1 | = 4 + 1 + 4 = 9,

which indeed equals 32.

[edit] External link