4. In the cells of a $3 \times 3$ square, the numbers $0,1,2, \ldots, 8$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is 18?
Answer: 2.

Solution. Let's color the cells in a checkerboard pattern: let the corner and central cells be black, and the rest white. From the condition, it follows that cells of different colors contain numbers of different parity. Since there are five black cells and four white cells, we get that the even numbers are in the black cells. Their total sum is \(0+2+4+6+8=20\). Therefore, the number in the central cell is \(2=20-18\).

Evaluation. 13 points for a correct solution. If an example of the arrangement of numbers satisfying the problem's condition is provided but the uniqueness of the answer is not proven, 6 points.

## Multidisciplinary Engineering Olympiad "Star" in Natural Sciences

## Final Stage

 2017-2018 academic year
## Problems, Answers, and Evaluation Criteria

## 6th Grade Variant 2

physics