Solution
The Broken Step
Show the problem again
A staircase has 12 steps. You start at the bottom (step 0) and must finish on step 12, and each stride carries you up exactly 1 or exactly 2 steps. Step 7 is broken: no stride may end on step 7. How many different climbs (sequences of strides) are possible? Give an integer.
Worked solution
**Answer: 65.** Since you can never stand on step 7 and a stride covers at most 2 steps, every climb must stand on step 6 and then take a 2-stride straight to step 8. So the climb splits into three independent parts: from 0 to 6, the forced jump 6 → 8, and from 8 to 12. Climbs of length n with strides of 1 and 2 are counted by Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 for n = 0, ..., 6. There are 13 ways to reach step 6 and 5 ways to go the 4 steps from 8 to 12, giving 13 × 1 × 5 = 65.
Source: Original problem written for AxiomIQ.