Answer
Method Used: Proof by Contradiction (Reductio Ad Absurdum)
Work Step by Step
To prove that there exists at least one day of the week on which 10 or more dates fall.
Hypothesis: Let us assume, on the contrary, that there is no day such that more than or equal to 10 dates fall on that day. Therefore each day of the week can be assigned at most 9 dates. But there are only 7 days in a week. So we can account for at most $7 \times 9 = 63$ dates.
But that contradicts the given fact that there are 64 dates.
Hence our hypothesis is absurd and there is at least one day of the week which is assigned greater than or equal to 10 dates.
