Bathhouse manager Mr. Sluníčko bought 58 umbrellas for the spa guests. Some were red and some were yellow. The red ones were packed in boxes of nine, while the yellow ones were in boxes of four. He bought both types of umbrellas in whole boxes. How many yellow umbrellas could there have been?

(L. Hozová)
We will proceed with the table as follows: We start by choosing the number of boxes with red suns, i.e., any whole number between 0 and 6 (if there were 7 or more boxes, there would be 63 or more red suns). Next, we determine how many red suns are in these boxes, i.e., nine times the number of boxes. Then we calculate how many suns will be yellow, i.e., subtract the number of red suns from 58. Finally, we determine how many boxes with yellow suns there should be, i.e., divide the number of yellow suns by four. In case the number of yellow suns is not a multiple of four, so that it cannot be divided by four without a remainder, we will strike through the relevant cell, and this option does not lead to a solution.

| boxes with red suns | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| red suns | 0 | 9 | 18 | 27 | 36 | 45 | 54 |
| yellow suns | 58 | 49 | $\mathbf{40}$ | 31 | 22 | 13 | $\mathbf{4}$ |
| boxes with yellow suns | - | - | 10 | - | - | - | 1 |

We get two possibilities: There could have been 40 yellow suns or there could have been 4.

Note. We can also proceed by choosing the number of boxes with yellow suns and then calculating the number of boxes with red suns. In this case, however, we need to consider all possibilities between 0 and $14(15 \cdot 4=60>58)$. We get the following table:

| boxes with yellow suns | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| yellow suns | 0 | $\mathbf{4}$ | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | $\mathbf{40}$ | 44 | 48 | 52 | 56 |
| red suns | 58 | 54 | 50 | 46 | 42 | 38 | 34 | 30 | 26 | 22 | 18 | 14 | 10 | 6 | 2 |
| boxes with red suns | - | 6 | - | - | - | - | - | - | - | - | 2 | - | - | - | - |

Evaluation. 1 point for explaining the procedure for one chosen (any) number of boxes with red (or yellow) suns; 3 points for correctly setting up and completing the table or an equivalent notation; 2 points for finding both possible numbers of suns.

If a solver is missing a column in the table due to inattention but otherwise their solution is correct and explained, award them 5 points. A complete procedure but without a verbal comment, evaluate at 5 points. If a solver stops listing columns of the table after finding the first solution, award them no more than 4 points. If a solver finds one solution by chance and does not provide a procedure or verification, award them 1 point; 2 points if they verify their result.