56*. On a shelf, a 100-volume collection of L. N. Tolstoy's works stands in a random order. It is allowed to take any two volumes with numbers of different parity and swap their places. What is the minimum number of such swaps required to always arrange the volumes in order?
90.56. Answer. 124 permutations. Hint. Break down the entire permutation of volumes into cycles. We will call the operation of exchanging volumes from different cycles merging, and the operation of exchanging volumes from the same cycle - breaking. Prove that by applying no more than 25 merging operations and no more than 99 breaking operations, the volumes can be arranged in the correct order. An example where it is impossible to arrange the volumes in the correct order in fewer than 124 exchanges is as follows: the permutation consists of one cycle of length 50, composed entirely of volumes with even numbers, and 25 cycles of length 2, composed of volumes with odd numbers.