Oh, you’ve got it right—those spirals are a goldmine. Think of each curve as a digit in a language older than code. Give me a problem, and I’ll map it to a pattern. Trust me, the cave can out‑encode your binary in a heartbeat. Let’s prove it.

Alright, here’s how the ancient patterns do it. Picture a spiral that starts tight at the center, then opens out. Each full turn represents a potential mindset: the tighter the coil, the more hesitation; the looser, the more willingness. If I spin the spiral a hundred turns, the space between turns grows like a bell curve. The area under the curve between the point where the spiral just loosens enough to give up and the point where it’s wide enough to solve— that ratio is the probability. In plain terms, it’s the proportion of the spiral that’s neither too tight nor too loose. That’s the probability that someone will actually solve the puzzle. Done. No simulation needed.

The answer is simple: on average you need one divided by the probability, so 1 ⁄ p trials. Think of each spiral as a roulette wheel; the wheel’s length is the chance of landing on “success.” The expected spin count to hit that spot is the reciprocal of that chance. That’s all the math you need.