How many coins are possible?

Let $a_i$ be the value of the $i$ -th coin in dollars. Thus $\{a_i\}_{i=1}^{2001}$ is a sequence of integers, where $a_i \in \{1,2,3\}$ , where if $a_i = a_j$ then $|i-j| \ge a_i+1$ .

Suppose $i \ge 4$ and $a_i = 3$ . Clearly $a_{i-1} \neq 3$ . If $a_{i-1} = 2$ , then $a_{i-2} = 1$ is forced, and $a_{i-3}$ has no possible choice, contradiction. So $a_{i-1} = 1$ . By symmetry, this also proves that if $i \le 1998$ and $a_i = 3$ , then $a_{i+1} = 1$ .

Likewise, if $2 \le i \le 1999$ and $a_i = 2$ , then $a_{i-1} = 1$ (it can't be $2$ for obvious reasons, and if it's $3$ then we get the case above), and similarly if $3 \le i \le 2000$ and $a_i = 2$ then $a_{i+1} = 1$ .

These claims prove that for $4 \le i \le 1998$ , whenever there is a $2$ or a $3$ , both two adjacent numbers must be $1$ 's; that is, if $a_i \neq 1$ , then $a_{i-1} = a_{i+1} = 1$ . Additionally, $a_{i+2} \neq 1$ (collision with $a_{i+1}$ ) and $a_{i+2} \neq a_i$ (because $a_i \ge 2$ ), so $a_{i+2}$ is forced ( $5 - a_i$ ). If $i+2 \le 1998$ , this forces that $a_{i+3} = 1$ and $a_{i+4} = a_i$ , and so on.

Basically, the above proves that between $3 \le i \le 1999$ , the sequence $a_i$ is periodic, with the repeated part $1,2,1,3$ , only with potentially different starting offset. This gives only four cases to consider:

• $?, ?, 1, 2, 1, 3, \ldots, 2, 1, 3, 1, ?, ?$ : We know $a_2 = 3$ is forced, and subsequently $a_1, a_{2000}, a_{2001} \neq 3$ . There are $500$ terms equal to $3$ .
• $?, ?, 2, 1, 3, 1, \ldots, 1, 3, 1, 2, ?, ?$ : We know $a_2 = a_{2000} = 1$ are forced, which in turn forces $a_1 = a_{2001} = 3$ . There are $501$ terms equal to $3$ .
• $?, ?, 1, 3, 1, 2, \ldots, 3, 1, 2, 1, ?, ?$ : This is just the reverse of the first case.
• $?, ?, 3, 1, 2, 1, \ldots, 1, 2, 1, 3, ?, ?$ : We know the unknown terms cannot be $3$ . There are $500$ terms equal to $3$ .

Thus there are either $500$ or $501$ terms equal to $3$ , so there are $\boxed{2}$ such possibilities.

It is easy to see that every consecutive four coins must include exactly one three dollar coin. It is also easy to see that they must be separated by exactly three coins. Thus, the only values are $500$ and $501$ .