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Answer: 316, or 5 hours 16 minutes

The digits will next be all the same at 11:11

5:55 to 6 = 5 minutes
6 to 11 = 5 hours
11 to 11:11 = 11 minutes
5 hours + 5 minutes + 11 minutes = 5 hours 16 minutes or 5$\times$60 + 16 = 316 minutes



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