What is the most common digit?
Daft as it may sound, there is indeed such a thing, and the digit is 1.  It doesn't matter whether you're talking about the lengths of rivers (in miles, kilometres, cubits or any other measurement), the populations of cities, the item prices on your shopping receipts or whatever: in each random set of numbers 1 is the most common of all digits.  This is Benford's Law, which is now understood to a sophisticated extent, and is used by the United States Inland Revenue Service and others to check for non-standard (i.e. doctored) distributions on corporate returns etc.

How many ways can you arrange things?
For 3 things, you can put them in 1 × 2 × 3 different orders, like this:

• 1-2-3
• 1-3-2
• 2-1-3
• 2-3-1
• 3-1-2
• 3-2-1

The formula is 1 × 2 × 3 × 4 for four objects, and so on.  These are factorials, denoted by an exclamation mark e.g. factorial 4 = 4! 

If you're putting individual letters into addressed envelopes, and you wickedly want to know how many different ways you can put all the letters into the wrong envelopes, this is far more involved.   Just one example: with 5 letters, there are 44 ways of getting every single one wrong, leaving 75 ways of getting some of them wrong, and only one way of getting the job right. 

How do you calculate Pi?
Here is a routine in pseudo-code:

Initial Values of Variables
a = 1; d = 1; y = 1; z = 1;
b = ( 1 / sqrt(2) )
c = 0.25
Loop indefinitely
y = a
a = ( a + b ) / 2
b =  sqrt(b × y)
c = c - (z × (a - y)² )
z = z × 2
display (a + b)² / (c × 4)

How many colours are needed for a map?
Four.  This has been known since 1878, but was not proved until 1976.  The final proof was hundreds of pages long, took 1200 computer-hours of calculations, and is beyond the understanding of all but a handful of specialist mathematicians.

How do you construct a magic square?

What is the largest number?
A googol is 10¹⁰⁰ which is pretty big,  and a googolplex is 10^googol, which is even larger.  But the winner by billions of universes is Graham's Number, which dwarfs both of these, and for that matter any other quantity ever conceived.  Graham's number is the answer to a problem in Combinatorics (an abstruse branch of Maths dealing with sets) and although the algorithm for calculating it can be described, the number itself is so large that it cannot be written down in any existing notation at all - even using towers of exponents.  Why not?  Because there would not be enough material in the universe to do so.  This is a monster of a number.