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Each fruit can have at most two fruits next to it but each type of fruit must be next to three other types of fruit so there are at least two of every fruit. This means there are at least $8$ fruits in total. In fact $8$ are sufficient, as shown in the arrangement
OABPOBAP (O for Orange, P for Peach, A for Apple, B for Banana)


This problem is taken from the UKMT Mathematical Challenges.