To round to the nearest hundredth (two decimal places), look at the third decimal digit (thousandths):
- $\(4.16496 \rightarrow\)$ third digit $\(4(<5) \rightarrow 4.16\)$
- $\(4.16501 \rightarrow\)$ third digit $\(5(\geq 5) \rightarrow 4.17\)$
- $\(4.16849 \rightarrow\)$ third digit $\(8(\geq 5) \rightarrow 4.17\)$
- $\(4.17469 \rightarrow\)$ third digit $\(4(<5) \rightarrow 4.17\)$
- $\(4.17496 \rightarrow\)$ third digit $\(4(<5) \rightarrow 4.17\)$
- $\(4.17501 \rightarrow\)$ third digit $\(5(\geq 5) \rightarrow 4.18\)$

So the numbers that round to 4.17 are:
4.16501, 4.16849, 4.17469, 4.17496.