Once I heard about a prophet that used math to foresaw with great accuracy many events of the humanity. Today I oddly realized the time between falling drops after washing cups fit the inverse square law.
What are some unexpected things accurately predicted by math?
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The angle of the wake of a body moving steadily in deep water (e.g. a ship or a duck) is always $2\arcsin(1/3) \approx 38.9^\circ$.
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Look at Good–Turing rule, the two posts in Azimuth blog:
The puzzle itself:
Suppose you go into the jungles of Ecuador and start collecting butterflies. You count the number of butterflies of each different species that you find. You get a list of numbers, something like this:
14, 10, 8, 6, 2, 1, 1, 1
Puzzle: What is the chance that the next butterfly you find will belong to a new species?
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A curious application from Fred Hoyle : Use the the prevalence of life forms on earth to deduce a new resonance in the carbon-12 nucleus.
Here is an interesting article about Mathematical Fortune-Telling.