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.

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?

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.

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.