, 2019

Let $n$ be a positive integer. Tracy writes a list of 10 whole numbers between 1 and $n$ (inclusive). Each number in the list is either equal to, one less than, or one more than the number before it.

For example, when $n = 7$:

Her list could be $5, 5, 6, 7, 6, 6, 5, 5, 6, 6$ or $4, 4, 3, 2, 1, 1, 1, 1, 1, 1$.

Her list could not be $1, 3, 3, 4, 5, 5, 6, 7, 7, 7$ or $5, 6, 7, 8, 7, 6, 5, 5, 5, 5$.

(a) Suppose that $n = 3$. Stacey forms a list by copying Tracy's list, except that whenever Tracy writes a 1, Stacey writes a 3, and whenever Tracy writes a 3, Stacey writes a 1.

(i) Which lists could Tracy write that would cause her list to be the same as Stacey's?

(ii) Explain why Tracy can write as many lists that start $2, 2, 1$ as start $2, 2, 3$.

(b) For which $n$ between 1 and 10 (inclusive) is the number of lists that Tracy could write odd?