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2018-12-17 Advanced

         

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!

\(n\) 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 \(P_n\) that none of these \(n\) students sit next to their good friends?

Submit your answer as the limit \( \displaystyle \lim_{n\to\infty} P_n. \)


Inspiration

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, \[ \left( \frac{\dot a}{a} \right)^2 = H_0^2 \left( \frac{1-\Omega}{a^3}+\Omega \right), \] where \(\Omega\) is the fraction of dark energy in the Universe, \(H_0\) is the Hubble constant today, and \(a(t)\) is a dimensionless "size factor" selected so that its value today is \(a=1\) \(\big(\)for example, \(a(t)=R(t)/R_0\), the distance \(R\) between Earth and a distant galaxy at time \(t\) divided by the current distance\(\big).\)

Based on the solution of this equation, what was the scale factor \(a\) when the Universe was half of the age today?

Notes: Use \(\Omega=0.68\). The quantity \(1-\Omega=0.32\) represents the fraction of matter (regular and dark) in the Universe. The Hubble constant is \(H_0=\dfrac{1}{14\times 10^9\text{ years}}\). 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.

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 \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers. Enter your answer as \(a + b\).

Details and Assumptions:

  • Each round of Yahtzee consists of up to 3 tosses. In the first toss, you roll all 5 dice.
  • In the second toss, you may roll some or all of the 5 dice.
  • In the third toss, you may roll some or all of the 5 dice.
  • The probability to obtain a "Yahtzee" assumes that you follow the optimal strategy.
  • It would be acceptable to use a calculator or computer software to help with computations.

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 \[F = \frac \varphi {\varphi + \sqrt{a - b\varphi^2}},\] where \(\varphi = \frac {1+\sqrt 5}2\) is the golden ratio, and \(a,b\) are integers. What is \(a-b?\)

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