Alex visits the museum every 6th day.
So, after today the days on which he will visit the museum will be in multiples of 6, i.e. 6, 12, 18, 24, 30 and so on days.
Alex visits the museum on 3rd June.
And, there are 30 days in June.
So, dates on which he will visit the museum will be:
3rd June + 6 days = 9th June
3rd June + 12 days = 15th June
3rd June + 18 days = 21st June
3rd June + 24 days = 27th June
Hence Alex visits on 3rd June, 9th, 15th, 21st, and 27th June.