## Spot It / Dobble

There are 55 cards in the game Spot it (sold in Europe as Dobble), with 8 symbols on each card and each two cards having exactly one symbol in common.

In the instructions, you will read that you will find over 50 different symbols in the game. Prove that there must be indeed a little more of these symbols.

• #### Hint

Consider the number of occurrences of a symbol.

• #### Solution

If each symbol has at most 7 occurrences, then there could be at most 49 cards in the game: For the selected card, each other must have at least one symbol in common, which gives $$1 + 8 {\cdot}6$$ cards (selected card + the number symbols on it * the number of the remaining occurrences).

Therefore, some symbol occurs at least eight times. If we take 8 cards with this symbol, the other symbols on these eight cards must already be different. Therefore, the total number of symbols is at least $$1 + 7 {\cdot}8 = 57$$.

Note: The cards in the game correspond to the selected 55 lines (out of a total of 57) of the projective plane of order 7, and indeed just 57 symbols are used.