The Broken Stick Problem.

You've got a stick. Two points are chosen independently and at a random on the stick. The stick is broken at those points to form three smaller sticks. What is the probability these three sticks can form a triangle?

Apparently this problem has a pedigree that goes back to 1854 examination at Cambridge University.
When you're ready for a neat solution. Here.
A more traditional solution is here.
And MIT Blossoms also has a video and teacher resources here

Reversible Number Arithmetrick

Here's a nice fact from those NRICH folks again about reversible numbers.

Equal Averages

There are several different notions of average: the mean, the median, the mode and the range (see below for the definitions). If you work out each of these statistics for the set of numbers 2, 5, 5, 6, 7, you'll notice something interesting — they are all equal to 5!

Can you find other sets of five positive whole numbers where mean = median = mode = range?

How many sets of five positive whole numbers are there with mean = median = mode = range = 100?

This puzzle comes from the NRICH site.

Eggstreme

Awash with chocolate eggs and gift bags, confectionery delirium sets in and your mind starts to wander... Given you have 20 bags, what is the minimum number of eggs needed so that you have a different number of eggs in each bag? 
This puzzle and solution can be found here.
There is an interesting non-obvious (quite mischievous) solution which nicely illustrates what mathematicians can do when the rules are not overly defined.

A 13 step problem

When I go up my stairs in a hurry, I take some steps two-at-a-time. This morning, I climbed my thirteen steps in this pattern: 1, 2, 1, 2, 1, 1, 2, 1, 2. Yesterday the pattern was 1, 1, 1, 1, 1, 1, 2, 2, 1, 2. How many days do you think I can go before I have to repeat a pattern?

This is from the NRICH site.
and is referenced in this blog post from Math with bad Drawings.

Zombie Logic Puzzle

Finger counting (Reversal)

This problem comes through the Collaborative Mathematics community.
Count on your fingers and thumb in the following way:
Start on your thumb with 1; Index finger is 2, Middle Finger is 3, Ring Finger is 4, Pinky Finger is 5. Then turn around and keep counting, so, Ring finger is 6, Middle Finger is 7, Index is 8 and Thumb is  9. Keep going in this way, changing the direction of counting at the pinky and thumb.

The question is "WHICH FINGER ARE YOU ON WHEN YOU COUNT THE NUMBER 1000?"
You can see video solutions to the problem here.

Extension questions include:
In which direction are you counting (Thumb to Pinky or Pinky to Thumb) when you reach 1000?
How many times have you changed direction before reaching 1000?
How many times have you touched your middle finger before reaching 1000?
Is it possible to figure out which finger you would end on for any number? eg. 321, 457? 1 billion?

Suppose you started at Zero on your thumb, how would that change which finger you ended on?

Suppose one has polydactyly and has 6 digits on a hand. Counting in the same way, which finger will you end on? What about if one has fewer than 5 digits on a hand, say 3, which finger will you end on?

Order of operations

6÷2(2+1)= ?

This problem turned up on my Facebook feed and generated a nice little conversation. We googled it, we tried it out in Wolfram Alpha - same answer came back. Why does this problem (potentially) cause difficulty? What does it tell us about necessary and arbitrary parts of mathematics? What other equivalent expressions could be written?

Gauss Triangular Array Problem

This is from the Gauss (2012) competition.
Solution here.

After solving the problem look for other patterns in this triangular array, or ask other mathematically interesting questions. Write your own problem based on the design of this array (or one like it).  

Counting dots and squares

This was inspired by a similar problem in the Canadian Math Kangaroo Parent paper (2014).
You can see that paper here.

Identical Coloured Nets

Net Problem from UKMT Senior Math Challenge (2014)
Solution here.

Regular Star Polygon made of Trapezia

Angle/Polygon Problem from UKMT Senior Math Challenge (2014).
Solution here and extended solution together with note on original inspiration for problem here.

Tetrahedral Die Problem

Patterning Problem from the UKMT Junior Mathematics Challenge (2014).
Solution here, extended solution and suggested investigation here.

Quadrilateral Areas Problem

Area Problem from the UKMT Intermediate Mathematical Challenge (2015).
Solution here and extended solution here.