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The Sheriff’s Star: Arithmetic Solution

The Sheriff’s Star: Arithmetic Solution

If you haven’t already read the problem description for the Sheriff’s Star, you might want to do that before looking at solutions.

The arithmetic solution doesn’t use anything beyond division, and so it’s theoretically fine for 1849 gold miners. It would be tedious to do some of these calculations by hand (I’d use a calculator), but it can be done.

My mom (Karen Butler Easter) was the first one to present this solution to me; I didn’t find it on my own. She’s an excellent lateral thinker, and it shows in this solution: straightforward and lovely.

Arithmetic Solution Master 2013-11-27 02_r1_c1Start by marking a center point and then use the string (acting as a radius) with a pencil at the end of it. Mark out a circle. (You could also probably do this with a compass or a pair of dividers—it seems likely that a goldsmith might have both.)

Arithmetic Solution Master 2013-11-27 02_r2_c1We could certainly calculate circumference based on some numeric methods here, but it’s easy just to measure it, and that’s likely to be precise enough for our goldsmith. So we measure it with the string, marking the string to show the total length.

We measure that length and then divide it by n, which is the number of points we want the star to have. We call this new amount x.

Arithmetic Solution Master 2013-11-27 02_r5_c3Then we go back to mark the string up with equally-spaced points. If you have a calculator, you can do this by multiplication or by repeated addition; if not, you might just mark the length of x on a piece of paper and use that repeatedly to mark each segment.

This approach is similar to one that’s frequently used for laying out repeated measurements in woodworking: a story pole or scantillion.

Arithmetic Solution Master 2013-11-27 02_r6_c3 Having marked up the string with equal segments, we transfer it back onto the circle and mark those points. This avoids all kinds of fun math that you’ll find in the trigonometric solution.

Arithmetic Solution Master 2013-11-27 02_r7_c3At this point, you can form the star however you want. Some badges have deeper grooves between the points. How might you accomplish that?

There you go! Simple, easy, and not much calculation involved. Head back to the Sheriff’s Star page to see links to other solutions people have created.

Have you come up with a new solution? Leave a comment!

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