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.
by
Jason Dyer
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?
by
Danielle Scarano
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.
by
Rohit Gupta
\[\begin{align} \underbrace{\dfrac23 + \dfrac23 + \cdots + \dfrac23}_{\color{green}{A}\color{black} \text{ copies of } \frac23} &= \underbrace{\dfrac45 + \dfrac45 + \cdots + \dfrac45}_{\color{blue}{B}\color{black} \text{ copies of } \frac45} \\\\\\ 20 \leq \color{green}{A}\color{black}+\color{blue}{B}\color{black} &\leq 24\\\\ \color{green}{A}\color{black}+\color{blue}{B}\color{black} &=\, ? \end{align}\]
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?
by
Pi Han Goh
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.
by
Edward Teach