# Problems of the Week

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# 2018-02-26 Basic

If I have the 5 colored shapes shown that I can rotate, and I use each shape once, is it possible to place them so they fit perfectly in a $$5 \times 4$$ rectangle?

Hint: Notice the checkerboard pattern of the rectangle, and consider how the shapes would be colored if they followed the same pattern.

Suppose you have two identical plastic bowls: bowl A and bowl B. Bowl A is empty, while bowl B is filled with room temperature water. If you place an identical ice cube in each bowl, which bowl's ice cube will melt first?

Friends Stella and Robert each have a ball of the same size, except Stella's ball is made of steel, while Robert's is rubber. They are having a competition to see who can launch their ball farthest. If they each have a cannon capable of accelerating their ball to $$50\text{ m/s}$$ at the same angle of projection, who will win?

Details and Assumptions:

• The balls are shot through the air, but no wind is blowing.
• The distance is measured from the cannon to the point at which the ball first hits the ground.

\begin{align} \underbrace{\dfrac23 + \dfrac23 + \cdots + \dfrac23}_{\color{green}{A}\color{black} \text{ copies of } \frac23} &= \underbrace{\dfrac45 + \dfrac45 + \cdots + \dfrac45}_{\color{blue}{B}\color{black} \text{ copies of } \frac45} \\\\\\ 20 \leq \color{green}{A}\color{black}+\color{blue}{B}\color{black} &\leq 24\\\\ \color{green}{A}\color{black}+\color{blue}{B}\color{black} &=\, ? \end{align}

I have $$\color{green}{A}\color{black}$$ copies of $$\frac23$$ on the left-hand side of the equality, and $$\color{blue}{B}\color{black}$$ copies of $$\frac45$$ on the right. In total, I've written down between 20 and 24 fractions. Exactly how many fractions are there?

The birthday paradox is a surprising result of probability. Suppose you randomly chose 23 people and put them in a room. Then there would be a good chance $$\big($$greater than $$\frac{1}{2}\big)$$ that two of those people share a birthday (even though there are 365 days in the year).

What about birth seasons (spring, summer, fall, winter)? Suppose you randomly chose 3 people and put them in a room. Then is it true that there would be a greater than $$\frac{1}{2}$$ chance that two of them share a birth season?

Note: Birth seasons do not all have the exact same likelihood. However, their likelihoods are close enough that you can assume they are equal for this problem.

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