15. They want to give seven gift packages to seven children, one to each. They want to ensure that each package contains three different toys and that, no matter which two children are chosen, they receive at most one toy in common. What is the minimum number of distinct types of toys that need to be used?
15. The answer is 7. That it is possible to create seven packages with the required properties using seven types of games is demonstrated by the following grid, where A, B, C, D, E, F, and G indicate the types of games and the columns give the composition of the packages.

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

Moreover, six types of games are not sufficient. We prove this by contradiction. Suppose that A, B, C, D, E, and F are sufficient for seven packages. In total, we have twenty-one games, so there must be a type of game, say A, that appears in at least three different packages; the other games that appear in these three packages are six (two per three) and must all be of different types from each other and all different from A, which would then give us seven distinct types of games, i.e., a contradiction.