B5. The sum of the number of eyes on opposite sides of a die is always 7. Nine identical dice are glued together in a $3 \times 3$ square. This is done in such a way that two glued sides always have the same number of eyes. In the figure, you see the top view of the result, where the number of eyes is not shown for five dice.

What number of eyes should be on the place of the question mark?

![](https://cdn.mathpix.com/cropped/2024_04_17_8b0e5b2f470f23b328a2g-1.jpg?height=354&width=366&top_left_y=2327&top_left_x=1462)

## C-questions

For the C-questions, not only the answer is important; your reasoning and the way of solving it must also be clearly written down. Solve each C-question on a separate sheet of paper. Each correctly worked-out C-question is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper (per question!) as well.
B5. 3 The numbers on opposite sides of a die are called opposite. Together, these numbers always add up to 7. We consider two dice that are touching with the same numbers, but can still rotate relative to each other. Around the two dice, we then see the same numbers, but in reverse cyclic order.

Consider the situation where the numbers on the top of the two dice are opposite, say \(a\) and \(7-a\). This situation is illustrated in the left figure. The left die then has the numbers \(a, b, 7-a\), and \(7-b\) around it for some \(b\). The right die has the numbers in reverse cyclic order: \(7-a, b\), \(a\), and \(7-b\). We see that the dice must have the same numbers on the front and back, namely \(b\) and \(7-b\), respectively.

Conversely, if the two dice have the same numbers on the front (or back), the numbers on the top must be opposite.
![](https://cdn.mathpix.com/cropped/2024_04_17_ff229bb29d2323b894dcg-2.jpg?height=352&width=1170&top_left_y=886&top_left_x=450)

We apply this to the six dice in the first two columns of the \(3 \times 3\) table, see the right figure. The two dice in the third row show opposite numbers on the top, namely 2 and 5. The numbers \(c\) and \(d\) with which they touch the two dice in row 2 must therefore be the same. From this, it follows that the numbers \(e\) and \(f\) on the top are again opposite. The numbers \(g\) and \(h\) are therefore the same. Finally, the numbers visible on the top of the two dice in row 1 must again be opposite. We conclude that a 3 must be placed where the question mark is.

## C-problems