The first question is: If today is a Monday, what day will it be in 31 days' time?
This is quite simple. 31 days contain 4 weeks (28 days) and 3 days and 3 days onward from Monday is Thursday so in 31 days' time, it will be Thursday.
The second question is: If April 1st is a Thursday, what day will Christmas day be on?
Likewise, we use the same method despite us having to do more calculations. 30 days + 31 + 30 + 31 + 31+ 30 + 31 + 30 + 24 equals 268 days/7 = 38 weeks and 2 days and 2 days onwards from Thursday is... Saturday! (our answer).
The third question is: If Jan 1st is a Friday, and it is a leap year, what day will Jan 1st be on next year?
Now, we use our general knowledge - 1 leap year has 366 days so 366/7 = 52 weeks and 2 days, and once again... 2 days onward from Friday is Sunday (our answer).
Now for the last question: Can you use today's date to work out what day of the week your birthday will be on?
I don't know about YOU but my birthday is on the 21st of July and today is the 28th of April so, using the same method (I won't go through the method), we will conclude that this year my birthday will be on a Wednesday.
Thank you. I hope this clarifies the question's solution! :-)