# Problems of the Week

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# 2018-01-01 Basic

A block has edge lengths of 3 cm, 4 cm, and 5 cm (as shown), where opposite faces have the same color. If the block is rolled, which color is most likely to come up on top?

Assume the block's mass is evenly distributed.

People use magnets to hang notes on refrigerators. Despite the downward pull of gravity, magnets do not fall, even though the magnetic force doesn't pull up.

How is this possible?

Camila claims that she's constructed a quadrilateral where, given the coloring below, the orange edges are all the same length, the blue edges are all the same length, and the orange edges are not the same length as the blue edges.

Clearly, Camila didn't accomplish what she set out to do. However, is a quadrilateral with the property stated above possible?

Determine the value of the expression below: $\frac{(6! + 5!) \times (4! + 3!) \times (2! + 1!)}{(6! - 5!) \times (4! - 3!) \times (2! - 1!)}.$

Note: The "$!$" symbol is the factorial operation, but there's a clever shortcut for solving this that entirely avoids multiplying out the factorials!

Santa has 5 elves in a line named Angel, Buster, Cinnamon, Dash, and Evergreen (designated A, B, C, D, and E) who were each assigned to wrap a present 1, 2, 3, 4, and 5. They each truthfully state the following:

A: "I wrapped a present with a number less than or equal to 3."
B: "I wrapped an even-numbered present."
C: "The elf that wrapped my assigned present was working next to me."
D: "Santa assigned Angel to wrap present 1, Buster to wrap present 2, Cinnamon to wrap present 3, me to wrap present 4, and Evergreen to wrap present 5."
E: "Teehee! None of us wrapped the present we were assigned!"

Each elf stands behind the present that he or she wrapped. Santa has to deliver present 1 to a child. Which elf's present should he deliver?

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