## 1100.11 – The Old Wooden Disc

The old wooden disc in the figure is about an inch thick and has sat for over a hundred years on the counter in a little country store in North Carolina. The circumference of the disc is exactly twenty-seven centimeters. The disc is marked on its edge in six places, two of which are shown in the diagram. The disc can be used to measure lengths of ribbon. The pegs are arranged so that any length of ribbon, from one to twenty-seven centimeters, can be measured by laying the ribbon between one pair of marks or another. Where should you make the other four marks so that this is possible?

## Solution

A key to solving this problem is to realize that, if you can find a way to measure $x$ centimeters of ribbon, you can also measure $27-x$ centimeters by wrapping the ribbon around the other side of the disc. This cuts the number of lengths that need to be checked nearly in half. There are multiple solutions to this problem. Here is one: