An international committee consists of 5 members, each representing 1 of 5 states. The documents that the committee works on are stored in an armored safe. How many locks should the safe have, and how many keys should each committee member be provided with, so that access to the documents is only possible when at least three, randomly chosen members of the committee are together?
a) Any two members of the committee cannot open the cabinet in the absence of the other three; therefore, for any pair of members, there must be at least one lock that neither of them has a key to; we call such locks the unopenable locks for the respective pair.

Two different pairs of members cannot have a common unopenable lock, because otherwise the common unopenable lock could not be opened even if the two pairs were present together, which would include at least three members of the committee.

Therefore, on the one hand, there must be at least as many locks on the cabinet as there are different pairs that can be chosen from the committee members, which is 10.

On the other hand, each committee member must be provided with at least as many keys as there are different pairs that can be formed from the other four members - that is, at least 6 keys - because each member must have a key to each of the 6 different unopenable locks corresponding to the 6 pairs that can be formed from the other four.

b) We will show that 10 locks are sufficient to meet the condition. Thus, we can assign only one unopenable lock for each pair. Any member of the committee forms a pair with each of the other four, so they do not get a key to the four unopenable locks corresponding to the appropriate pairs, but they do get a key to the other 6. From the point of view of the locks, only the two members who form the pair for which a given lock is unopenable do not get a key to that lock, the other three do, so in the presence of any three members of the committee, at least one key is available for every lock.

![](https://cdn.mathpix.com/cropped/2024_05_02_1c8462509a6d209a2955g-1.jpg?height=420&width=458&top_left_y=844&top_left_x=820)

1. $\quad 2 . \quad 3 . \quad 4 . \quad 5 . \quad 6 . \quad 7 . \quad 8 . \quad 9 . \quad 10$.

| $\mathrm{A}$ | - |  |  |  | - | - |  |  | - |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\mathrm{B}$ | - | - |  |  |  |  | - |  |  | - |
| $\mathrm{C}$ |  | - | - |  |  | - |  | - |  |  |
| $\mathrm{D}$ |  |  | - | - |  |  | - |  | - |  |
| $\mathrm{E}$ |  |  |  | - | - |  |  | - |  | - |

In the diagram and table, $A, B, C, D, E$ represent the committee members, and $1,2, \ldots, 10$ represent the locks and their keys. The number next to the line connecting the points $A, B, C, D, E$ in pairs indicates the unopenable lock for that pair. In the table, a ,,-" symbol indicates that the person corresponding to the row does not have a key to the lock corresponding to the column. Each column has three unmarked cells.

Iren Solyom (Budapest, Veres Pálné g. III. o. t.)