On each wall of the dice, we wrote a different prime number less than 20 so that the sums of two numbers on opposite walls were always the same. We placed the dice on the first square of the plan with the smallest number facing down. Then we flipped the dice in the indicated direction along the plan. Each time the dice touched the plan, we wrote the number it touched on the corresponding square. Which number did the dice touch on the shaded square if the sum of all the written numbers was

1. the smallest possible,
2. the largest possible?

(The plan consists of squares that are the same size as the walls of the dice, see fig.)

![](https://cdn.mathpix.com/cropped/2024_04_17_4fe13a169d972f69d5ebg-1.jpg?height=649&width=491&top_left_y=1463&top_left_x=1388)

(M. Dillingerová, L. Hozová)
Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. We are looking for three pairs of these prime numbers that have the same sum, which are the pairs $(11,13),(7,17)$, and $(5,19)$ with a sum of 24. These pairs represent the numbers on opposite faces of the die. According to the problem, we wrote the number 5 on the first square of the plan. The image marks pairs of squares that were touched by opposite faces of the die.

No matter how we roll the die, the sum of all the numbers written on the plan will be the same, specifically $24 \cdot 8=192$. In both cases, the die could have touched the colored square

![](https://cdn.mathpix.com/cropped/2024_04_17_4fe13a169d972f69d5ebg-2.jpg?height=643&width=483&top_left_y=244&top_left_x=821)

with any of the numbers 7, 11, 17, or 13, i.e., with any number except 19, which lies on the opposite face of 5.

Scoring. 1 point for the pairs of prime numbers; 3 points for the consideration of the sum of the written numbers; 2 points for the conclusion and answer.