1. Albert doesn’t know the month because it could be one of 4 possible options

2. Bernard does not know the day because it could be one of 6 possible options

3. However, if Bernard had been told 18 or 19 then he would know the date because there is only one option for each

4. The fact that he doesn’t means that both men can rule out June 18 or May 19

5. This leaves two possible options for May

6. One for June

7. Two for July

8. And three for August

9. We can now deduce that both men realise Albert could not have been given June, otherwise he would now know the date because there is only one possible option. We can therefore rule out June 17

10. This leaves us with two possible options in May

11. Two in July

12. And three in August

13. However, it is Bernard who now realises the answer, so his number must reveal the date

14. If he had been given 14, 15 or 16 he would still not know because there are still two options for each

15. Albert realises this

16. Therefore the only way that both men can deduce the date is if Bernard was told 17

because there is only one remaining option for 17

17. Thus the solution to the problem is August 17