Maths Puzzle - Digits, Dates, Clock

What is the last digit in 

• 11**11
• 4**65
• 6**15
• 7**23
• 18**23
• 19*101
• Which is bigger 9**25 or 25**9

And clocks:

• At what time between 4PM and 5PM are the minute and hour hands together?
• At what time between 3PM and 4PM are the two hands 15 deg from each other

Start with 12 O Clock position as the zero degree. At x minutes past 4PM what are the positions of the hour hand and minute hands in degrees from 12 O Clock?

Calendar: https://docs.google.com/spreadsheets/d/19aRJBpZz8O2TY1nbMsSoTm2RycN3SqT9qH_s1VZlCYs/edit#gid=612883050 .
(the file has one error. February is hard coded to have 28 days.. It doesn't consider leap years. To that extent the day of the week calculated will be off by 1 day for dates in a leap year after the month of Feb)

What day of the week was

• Jan 29, 1934
• Feb 1, 2217
• Oct 2, 833

Solution tip:

• Every year (normal one) has 52 week + 1 odd day. (I came across this terminology in a maths book) and am using the same terminology here.
• Every leap year has 52 weeks + 2 odd  days
• Every 4 years has 3 normal years + 1 leap year and hence 3*1+1*2=5 odd days total. so if today (Nov 21, 2017) is a Tue, then the day 4 years later Nov 21, 2021 has 5 odd days and hence 5 days after Tue. Count  Wed=1 odd day, Thu=2, Fri=3, Sat=4, Sun=5. Hence that day is a Sunday
• Every 100 years has 24 leap years and 76 normal years because the century year like 2100, 2200 are not leap years. But century years such as 2000, 1200, 400, 3200 etc are leap years. Don't ask me why. That's how it is. Hence 100 years have 24*2+76*1=124 days = 17 weeks and 5 days. Removing the exact number of weeks, the odd days in every 100 years = 5.
• Every 400 years, number of odd days = 3*5+1*6=21 = 3 weeks exactly. Hence odd days = zero. 400 years has three numbers of hundred years (which are not leap years) and one number of 400 years (which is a leap year)
• Let's take Nov 21, 2017. We count the following to arrive at this date
• 400*5=2000 years
• 4*4=16 years
• 1*0=0 years
• 10 months (Jan to Oct of a non leap year)
• 21 days
• Now, 
• 5 sets of 400 years give 5*0=0 odd days 
• There are no sets of 100 years - so 0 odd days
• 4 sets of 4 years give 4*5=20 odd days
• There are no sets of 1 years - so 0 odd days
• Jan till Oct: 365 - (31+30) =304. I subtracted Dec and Nov days from 365 which is easier than counting all days from Jan to Oct. 304 days gives 43 weeks and 3 odd days
• 21 days in Nov=3 weeks and 0 odd days
• Counting only odd days: 0+0+20+0+3+0=23 = 3 weeks and 2 odd days = 2 odd days. 
• Now the final step is this: take the number from the previous step which is two in this case. If zero then Sun, if one then Mon, if two then Tue and so on. Hence Nov 21, 2017 is a Tue.

What is the date 121 Tuesdays before Oct 29, 1965?