Discussion: This student immediately recognizes the recursive nature of the problem by analyzing what happens at the end of a the row: either the child there seats back in their seat or the last two children must swtich seats. This reduces the number of ways, say C_n, to the sum of C_n-1 and C_n-2 and so the number of ways satisfies the Fibonacci numbers. Sine the first two values are 1 and 2, thereafter all the numbers coincide with the Fibonacci numbers. This is proved by a well executed induction.

Written response: Good execution of induction. Good job of recognizing how to recursively relate C_n to C_n-1 and C_n-2.