I believe that instead of “Should you pay $1 for this box?”, the question should be “Should you pay $1 for each use of this box?”. Otherwise, I believe I can just cut the island in half, go back and forth between the sides a billion billion times and keep pressing the button and look for bias in the responses. Then throw away the worse half of the island and repeat
Not a billion billion times. You need ≈2^100 presses to get any signal, and ≈O(2^200) presses to figure out which way the signal goes. 2^200≈10^60. Planck time’s about 10^-45 seconds. If you try to press the button more than 10^45 times per second the radiation the electrons in the button will emit will be so high frequency an small wavelength that it will collapse into a black hole.
I believe that instead of “Should you pay $1 for this box?”, the question should be “Should you pay $1 for each use of this box?”. Otherwise, I believe I can just cut the island in half, go back and forth between the sides a billion billion times and keep pressing the button and look for bias in the responses. Then throw away the worse half of the island and repeat
Not a billion billion times. You need ≈2^100 presses to get any signal, and ≈O(2^200) presses to figure out which way the signal goes. 2^200≈10^60. Planck time’s about 10^-45 seconds. If you try to press the button more than 10^45 times per second the radiation the electrons in the button will emit will be so high frequency an small wavelength that it will collapse into a black hole.