not a Bernoulli factory

May 19, 2020
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A Riddler riddle I possibly misunderstood:

Four isolated persons are given four fair coins, which can be either flipped once or returned without being flipped. If all flipped coins come up heads, the team wins! Else, if any comes up tails, or if no flip at all is done, it looses. Each person is further given an independent U(0,1) realisation. What is the best strategy?

Since the players are separated, I would presume the same procedure is used by all. Meaning that a coin is tossed with probability p, ie if the uniform is less than p, and untouched otherwise. The probability of winning is then

4(1-p)³p½+6(1-p)³p½²+4(1-p)p³½³+p⁴½⁴

which is maximum for p=0.3420391, with a winning probability of 0.2848424.

And an extra puzzle for free:

solve xxxx=2020

Where the integral part is the integer immediately below x. Puzzle that I first fail solving by brute force, because I did not look at negative x’s… Since the fourth root of 2020 is between 6 and 7, the solution is either x=6+ε or x=-7+ε, with ε in (0,1). The puzzle then becomes either

(6+ε)⌊(6+ε)⌊(6+ε)⌊6+ε⌋⌋ = (6+ε)⌊(6+ε)⌊36+6ε⌋⌋ = (6+ε)⌊(6+ε)(36+⌊6ε⌋)⌋ = 2020

where there are 6 possible integer values for ⌊6ε⌋, with only ⌊6ε⌋=5 being possible, turning the equation into

(6+ε)⌊41(6+ε) = (6+ε)(246+⌊41ε) = 2020

where again only ⌊42ε=40 being possible, ending up with

1716+286ε = 2020

which has no solution in (0,1). In the second case

(-7+ε)(-7+ε)(-7+ε)-7+ε⌋⌋ = (-7+ε)(-7+ε)(49+-7ε⌋)= 2020

shows that only -7ε=-3 is possible, leading to

(-7+ε)⌊46(-7+ε)) = (-7+ε) (-322+46ε⌋)=2020

with only 46ε=17 possible, hence

2135-305ε=2020

and

ε=115/305.

A brute force simulated annealing resolution returns x=-6.622706 after 10⁸ iterations. A more interesting question is to figure out the discontinuity points of the function

ℵ(x) = xxxx

as they seem to be numerous:

For instance, only 854 of the first 2020 integers enjoy a solution to ℵ(x)=n.

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