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It is clear that there is a unique way to complete the top three rows, as shown (start in the second square of the third row). Thereafter it is possible to complete the fourth row with R and S alternating and the fifth row QPQPQ.

Therefore the shaded square can be either an R or an S.
 
 
 
 
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.