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There are two chests. One will open if you say a true statement and the other will open if you say a false statement, but you don't know which is which!
You also know that one contains treasure and the other will release a deadly gas, but again, you don't know which.
Is it possible to make a statement that will cause only the chest with treasure to open?
Hint: Each statement is either true or false. It is not possible for a statement to be a paradox!
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n students are arranged in a line such that each student is sitting next to their good friends, and each student has no other good friends other than the students they are sitting next to.
If a teacher randomly reallocates their seats, what is the probability Pn that none of these n students sit next to their good friends?
Submit your answer as the limit n→∞limPn.
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The Friedmann equation, derived from Einstein's theory of general relativity, describes the expansion of the Universe after the Big Bang. The equation is, in a slightly simplified form, (aa˙)2=H02(a31−Ω+Ω), where Ω is the fraction of dark energy in the Universe, H0 is the Hubble constant today, and a(t) is a dimensionless "size factor" selected so that its value today is a=1 (for example, a(t)=R(t)/R0, the distance R between Earth and a distant galaxy at time t divided by the current distance).
Based on the solution of this equation, what was the scale factor a when the Universe was half of the age today?
Notes: Use Ω=0.68. The quantity 1−Ω=0.32 represents the fraction of matter (regular and dark) in the Universe. The Hubble constant is H0=14×109 years1. We neglected the energy contained by radiation and assumed that the Universe is flat, i.e. the matter and energy fractions add up to 1.
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I play a round of the game "Yahtzee!", and my goal is to obtain the namesake combination: 5 dice all showing the same number.
What is the probability that I will be able to accomplish this?
The probability can be expressed as ba, where a and b are coprime positive integers. Enter your answer as a+b.
Details and Assumptions:
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The surface of a soccer ball is covered with pentagons and hexagons in such a way that one pentagon and two hexagons meet at each vertex.
Now, more mathematically, let's assume that a soccer ball is a truncated icosahedron with 12 identical regular pentagons and 20 identical regular hexagons.
The fraction of the pentagonal area on the surface of the polyhedral soccer ball can be expressed as F=φ+a−bφ2φ, where φ=21+5 is the golden ratio, and a,b are integers. What is a−b?
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