Four players, $A, B, C$, and $D$ are playing dominoes. Their set consists of 28 dominoes. The players take turns in the above order, and they can start with any tile. After two rounds, the sum of the numbers on the tiles placed by each player is: $A: 23, B: 20, C: 18, D$: 16. Then $A$ placed the 6-2 tile. What were the first 8 dominoes placed and in what order? (The domino set includes all possible combinations of numbers from 0 to 6: The tiles are placed in the usual manner, with any matching end being placed next to it.)
Player $A$ could only score 23 points by placing the two highest, the 12-point 6-6 and the 11-point 6-5 dominoes. Among the remaining tiles, there are two 10-point, two 9-point, and three 8-point tiles, but the latter, the 6-2, was placed on the table in the third round. Thus, $B$ could only score 20 points with the 10-point $6-4$ and $5-5$, $C$ could only score 18 points with the $6-3$ and $5-4$, and $D$ could only score 16 points with the two 8-point $5-3$ and $4-4$ tiles.

The eight dominoes placed in the first two rounds are therefore as follows: $A$: $6-6, 6-5$; $B$: $6-4, 5-5$; $C$: $6-3, 5-4$; $D$: $5-3, 4-4$. To determine the order of placement, note that among the sixteen numbers listed, the 6 and the 5 appear an odd number of times, so the chain of eight dominoes must end in 6 on one side and 5 on the other.

The eighth domino was placed by $D$ at one end of the chain, so it could only be the $5-3$, connecting to the only other 3, the $6-3$. If $C$ had placed this domino in the first round, then the first six placed dominoes - 6-6; 6-5; 6-4; 5-5; 6-3; 4-4 - would all have the four numbers appearing an odd number of times, which is not possible.

The last two dominoes placed were therefore the 6-3 and the 3-5; that is, after the sixth domino was placed, a 6 stood at both ends of the chain. $C$ could then place the second tile at either end, and $B$ placed the 6-4 domino in the second round. In this case, $B$ could only have placed the 5-5 in the first round if $A$ started with the 6-5.

The dominoes, therefore, could only have been placed on the table in the following order: $A$: 6-5, $B$: 5-5, $C$: 5-4, $D$: 4-4, $A$: 6-6, $B$: 4-6, $C$: 6-3, $D$: 5-3. It can be seen that the found order is possible according to the rules of the game. We also saw that in this order, the arrangement of the dominoes can be in two ways, depending on which side $C$ places the 6-3 tile in the second placement.

1986-01-025-1.eps

$1986-01-025-2 . e p s$

Note. Regarding the second part of the problem, many solutions were limited to simply stating the correct order. Given the nature of the problem, the solution can also be found through longer or shorter trial and error, which can result in a "fishing" style of writing. We would like to emphasize again that such solutions are fundamentally considered incomplete.