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Answer: 8


Each fruit type needs to touch all of the other three types, but each fruit can only be next to two other fruits. 

So there must be at least 2 fruits of every type $\Rightarrow$ at least 8 fruits altogether.

Find an arrangement with 8 fruits to show that this is possible:
 
OABPOBAP (O for Orange, P for Peach, A for Apple, B for Banana).

 

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.