Chris has an old 6-inch ruler that is missing the \(4 \text{ in}\) mark. However, he can still measure all integer lengths from \(1 \text{ in}\) to \(6 \text{ in}.\) He'd like to see if he can erase more marks and still be able to measure all integer lengths from \(1 \text{ in}\) to \(6 \text{ in}.\)

**What is the minimum number of marks the ruler could have?**

Note that all measurements must be made directly using the ends of the ruler and/or its marks. For example, you can't repeatedly use a distance of \(1 \text{ in}\) to make larger distances.

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