3. The numbers $1,2,3,4,5,6,7,8,9$ are written into the cells of a $3 \times 3$ table. After that, all possible sums of numbers standing in adjacent (by side) cells are written down in a notebook. What is the smallest number of different numbers that could have been written in the notebook?
# Solution:

Consider the number in the central cell of the table. Next to it are 4 different neighbors - they give 4 different sums with the central number, so there are already at least 4 different sums written down. An example where there are exactly 4 of them exists (one is shown on the right, the sums are $8,9,10,11$).

## Criteria:

There is an example with 4 different sums (to be checked!) - 2 points.

There is an estimate - that is, an explanation of why it is impossible to have fewer than 4 different sums - 3 points. Both a correct estimate and a correct example - 7 points.