OBM 1989 Problem 4

A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second.

At the $n$ -th turn, $n=1,2,\ldots,10$ , A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A.

B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins.

[list=a] [] Which player has a winning strategy? [] Suppose now both players has $k$ chips numbered 1 to $k$ . Which player has a winning strategy? [*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? [/list]