Reasoning with sticks

 

One of the recurring patterns in my number theory study has been getting trapped in circular reasoning without noticing it. I read an explanation, restate it in my own words, and feel as if I have understood something. But when I try to justify the idea, the entire argument often turns out to be feeding itself.

This happened most clearly when I was trying to understand why Euclid’s Algorithm works. Textbooks say, “Replacing a pair $(a,b)$ with $(b, \mathrm{a }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>b)$ does not change the gcd.” I kept trying to explain this using statements like “the gcd stays the same when you subtract multiples,” or “the algorithm preserves common divisors.” Both sounded correct, but both were really just the conclusion restated in slightly different language. Nothing in the reasoning actually began from a place that did not already assume the result.

The loop broke the moment I pictured the numbers as two sticks: one long stick of length $a$ and a shorter stick of length $b$. If you measure the longer stick using the shorter one, the piece left over after laying down the shorter stick repeatedly is exactly $\mathrm{a }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d </mi></mrow>b$. Any stick that measures both $a$ and $b$ will also measure the leftover piece, because that piece is created by repeating the shorter stick until it no longer fits. The idea becomes physical at that point. It is not an abstract rule but a direct observation about lengths.

The reverse direction also becomes obvious. If a stick divides $b$ and the leftover piece, then by assembling the leftover pieces and full pieces of $b$, it will divide $a$ as well. The equivalence is suddenly clear, and the reasoning does not run in a circle anymore. There is a concrete starting point that does not depend on the conclusion.

I had a similar experience with linear Diophantine equations. I knew the statement: the equation $ax+by=c$ has a solution exactly when $c$ is divisible by $\gcd (a,b)$. I tried to motivate it by saying that combinations of $a$ and $b$ somehow determine what is possible, and therefore the gcd must divide $c$. That felt intuitive but it was another loop. I was assuming the existence of the combination in order to justify the condition for the combination.

The picture that fixed it was again based on sticks. Consider all lengths you can create by cutting and taping together sticks of lengths $a$ and $b$ Those lengths do not appear randomly. They form a neat pattern, and the smallest positive one you can create is the gcd. Everything else is a multiple of that smallest value. If you want to create length $c$, it must fit into that pattern. If it does not, no rearrangement of the pieces will ever give you that exact length. This explanation begins somewhere real, so the loop disappears.

What I have learned is that circular reasoning often shows up when I keep everything inside symbols. Symbols can hide gaps because they feel precise even when the underlying logic is not. When I switch to a picture or an analogy, the reasoning becomes grounded, and the loop cannot sustain itself.

Now, when an explanation feels too smooth, I check whether it actually begins from a stable point. If it does not, I translate the idea into something I can visualize. Every time I do that, the reasoning stops circling and finally lands.

Also, here's a painting I made