Climbing the Stairs
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?
"Even if I only do a 2-step once, I can choose to take that step from any one of 12 places on the staircase. And I could have two, or three, or four 2-steps... so there will be lots more possibilities. It's going to be at least 50."
I wonder if there's a different pattern of climbing the stairs for each day of the year
How can you be sure you've included every possibility?
Is there a quick way to work out the number of different ways to go up staircases with many steps?
How can you be sure that your method will work for all staircases, not just the ones you've tried?
How many steps would I need on my staircase if I wanted to have more than a thousand ways of going upstairs?
What if I could also go down 3 steps at a time? Or 4 steps?
What if I could only go down 2 or 3 steps at a time?
The car park at work costs £1.50. The machine only accepts 10p and 20p coins.
I noticed that I have different ways of putting the money into the car park machine...
Yesterday I put in two 10p coins, then two 20p coins, then four 10p coins, then a 20p coin, and finally three more 10p coins.
This morning I put in four 10p coins, then three 20p coins, then another 10p coin, then two more 20p coins.
I think I could probably go for the rest of the month without feeding the machine in the same way twice.
Can I feed the machine in a different way each day of the year?
Can you think of some more manageable problems you could work on, to help you to get started?
What other interesting mathematical questions can you think of to explore next?
How can you be sure you've counted every possibility?
How would you work out the number of different ways of feeding the machine when the cost rises to £1.60? To £1.70? To...
How can you be sure that your method will work for all car park costs, not just the ones you've tried?
What if the car park machine also accepted 50p coins? Or 5p coins? Or...