We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes can range from 0 to 6, and the touching sides of the dominoes must have the same number of points.)

![](https://cdn.mathpix.com/cropped/2024_05_02_3b9ae63d491b27765f08g-1.jpg?height=222&width=355&top_left_y=317&top_left_x=880)
Solution. The dominoes with the smallest total point values are $0-0, 0-1, 0-2, 1-1, 0-3, 1-2, 2-2, 1-3, 0-4$. The first six of these have a total of 11 points, but it is not possible to form a closed chain with them, because the 3 only appears on the 0-3 domino. Therefore, the total must be at least 12 points. This is indeed achievable: using the dominoes 0-0, 0-1, 0-2, 1-1, 1-2, 2-2, the total points sum to 12, and the chain can be formed as shown in the diagram.