If these $10$ sentences are true, the nose will shorten $10$ cm. There are $(13-11+10)\div(5+1)=2$ sentences which are false. Thus, $10-2=8$ sentences are true. 
Pinocchio has a magic nose, which will stretch out $5$ cm if he tells a lie and will shorten itself $1$ cm if he tells a truth. At the beginning his nose was $11$ cm long. The length of his nose changed into $13$ cm after he had said $10$ sentences. If these sentences were not true, they were definitely false. Thus, there were~\uline{~~~~~~~~~~}~truths among these $10$ sentences. 
1: $$5$$ 
2: $$6$$ 
3: $$7$$ 
4: $$8$$ 
5: $$9$$ 