7. A die has different numbers of dots on each side, ranging from one to six, and the sum of the number of dots on opposite sides of the die is always 7. A cube of dimensions $10 \times 10 \times 10$ is formed from 1000 such dice. What is the smallest possible number of all dots visible on the sides of the resulting cube?

## Ministry of Science and Education of the Republic of Croatia  Agency for Education and Education  Croatian Mathematical Society

## SCHOOL/CITY COMPETITION IN MATHEMATICS  February 15, 2022.  5th grade - elementary school

In addition to the final result, the procedure is also graded. To receive full points, it is necessary to find all solutions and determine that there are no others, write down the procedure, and justify your conclusions. Problems 1. - 5. are scored with six points, and 6. and 7. with ten points.

## Problems worth 6 points:
First solution.

A cube has 8 vertices.

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-05.jpg?height=108&width=117&top_left_y=1154&top_left_x=227)

Each vertex determines one playing die.

The smallest number of visible dots on each such die is $1+2+3=6$.

Therefore, the smallest possible number of visible dots on all eight playing dice at the vertices is $8 \cdot 6=48$.

Each edge is determined by 8 dice that are not at the vertices.

The smallest number of visible dots on each such die is $1+2=3$.

Therefore, the smallest possible number of visible dots on all edges is $12 \cdot 8 \cdot 3=288$.

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-05.jpg?height=620&width=401&top_left_y=2037&top_left_x=242)
Each face is determined by $8 \cdot 8=64$ dice that are not on the edges. ..... 1 POINT
The smallest number of visible dots on each such die is 1.
Therefore, the smallest possible number of visible dots on all faces is $6 \cdot 64 \cdot 1=384$. ..... 1 POINT
The smallest possible number of visible dots on such a cube is $48+288+384=720$. ..... 1 POINT

##