Here’s a conjecture that’s been open for more than a hundred years. Prove that every closed curve in the plane has an inscribed square. To unpack that for non-math readers: suppose I draw some loop on a piece of paper, which might be really complicated, cross over itself, zig-zag back and forth a lot, doesn’t matter — as long as it eventually returns to its starting point. Then the problem is to show that, no matter how crazy the loop, you can always find four points on the loop which form a perfect square.
It’s a fun exercise to try to convince yourself that this is even plausible! Also, the version of the problem with “square” replaced by “equilateral triangle” is much easier.