# Problems of the Week

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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!

$$n$$ 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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