4. Alice has a number of cards. On each card, three of the letters A through I are written. For any two letters you choose, there is at least one card where those two letters appear together.

What is the smallest number of cards Alice can have?
B4. 12 There are 9 letters, namely $A$ through $I$. The number of possible pairs of letters is $\frac{9 \times 8}{2}=36$. Indeed, each of the 9 letters can form a pair with 8 other letters, but this way we count each pair twice (pair AE is the same as pair EA).

Each card has three different letters that form 3 pairs. Therefore, at least $\frac{36}{3}=12$ cards are needed to get each pair exactly once. That it can actually be done with 12 cards follows by taking the following triplets:

| ABC | DEF | GHI |
| :---: | :---: | :---: |
| ADG | BEH | CFI |
| AEI | CDH | BFG |
| AFH | BDI | CEG |

The triplets are also illustrated in the figure. The twelve triplets are formed by the three horizontal lines, the three vertical lines, the two diagonals, and the four curved lines. It is easy to verify that each pair of letters indeed lies on one of the lines.

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-4.jpg?height=574&width=577&top_left_y=981&top_left_x=748)