Here's a puzzle for you - based on the stupid things I think about when I'm bored.

How can you move from imprecise measurements to precise measurements?

Here's a story to go with it and help explain the question.

I'm cutting boards to lay out a deck, and I want it to look nice, so I want them to all be the same length. All I have is a handsaw. What do I do?

My solution is to cut the first board and use it as a measuring stick. In other words, I lay it on the next board and use it to strike a line. But, the first cut wasn't perfect (not perfectly square, not perfectly straight), I don't strike a perfect line, and my second cut isn't perfect. So, I can see some small differences from board to board.

A slight improvement is to use the first board as a "jig". I buy myself some clamps, clamp my "jig" to the board I want to cut, and use it to guide the saw.

That's better, but remember that my first cut wasn't perfect. I could buy a better jig, but now we get to the bottom line. How did the guy I buy it from make the jig? If he can do it, why can't I?

I'll say (for this puzzle) it's cheating to use a tool made by someone else. You have to start from scratch, using imprecise tools, and develop a method for improving precision.

I did some searching and didn't come up with an answer. So, I devised my own solution, but it seems a little cheesy. I'm curious to see what other people might come up. Maybe I'm missing something obvious.

Maybe my question was confusing ... or maybe just boring.

We always hear of measurements becoming more precise, and I simply wonder by what method they become more precise. How can inaccuracy create accuracy?

My thought was that a use of proportions would do it - like scissors, where I operate very long handles, and the output on the other side uses short blades. So, suppose the handles are 10 cm long and the blades are only 2 cm long. My current precision is a motion of 1 cm on the input. That produces a precision of 0.2 cm on the output. The same would be true of a gear pair.