6. A chess tournament was attended by 8 players, and each played one game against each other. $Z a$ victory earned a player 1 point, a draw half a point, and a loss no points. At the end of the tournament, all participants had different point totals. The player who finished in 2nd place scored the same number of points as the last four combined. Determine the result of the game between the 4th and 6th players in the overall standings.
SOLUTION. The last four players played 6 games among themselves, so the total number of points they scored together is at least 6. The player who finished in 2nd place, therefore, scored at least 6 points. If he had scored more than 6, i.e., at least 6.5 points, the best player (due to the condition of different scores) would have to have scored all 7 possible points; he would have defeated the player in 2nd place, who would thus have scored less than 6.5 points, which is a contradiction. The player in 2nd place, therefore, scored exactly 6 points. However, the last four players also scored exactly this many points in total, so they could only have earned these points from games against each other, which means they lost all their games against players in the first half of the final ranking. The player who finished in 6th place, therefore, lost the game against the player who finished in 4th place.

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