At the beginning, the applet displays 2N natural numbers: 1, 2, 3, ., 2N - 1, 2N. Select any N numbers by clicking on N numbers in turn. The selected numbers will be arranged in a column on the left in the decreasing order. The remaining N numbers will be arranged in the increasing order on the right.
Thus the set of numbers N2N = {1, 2, 3, ., 2N - 1, 2N} is split into two sequences, one decreasing and one increasing:
a1 > a2 > a3 > . > aN and
b1 < b2 < b3 < . < bN.
The applet is supposed to suggest a theorem due to V. Proizvolov, a well known Russian mathematician:
|a1 - b1| + |a2 - b2| + . + |aN - bN| = N2.
The proof is based on the observation that for every i, i = 1, ., N, one of the numbers in the pair ai, bi belongs to NN = {1, 2, ., N}, the other to NN+1, 2N = {N+1, N+2, ., 2N}.
Assume on the contrary, that, for example, that, for some i, both numbers in the pair belong to NN:
(1)
ai < N + 1
bi < N + 1.
This would imply that (together with ai) there are in the very least (N - i + 1) of the sequence {a} that are below N + 1. There also would be at least i terms from the second sequence {b} below N + 1. Together this would give N + 1 distinct natural numbers less than N + 1. But there are only N of them; thus our assumption (1) led to a contradiction.
The case
(2)
ai > N
bi > N.
leads to a contradiction in a similar manner. Thus indeed, in any pair ai, bi one number belongs to NN, the other to NN+1, 2N. This implies that
