# Problems of the Week

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# 2017-11-27 Basic

Trotter the Turkey wants to collect all of the acorns while avoiding the pesky squirrels.

Can Trotter do this without visiting any square twice?

Philip forms three equal portions of leftover mashed potatoes into three shapes: a cube, a sphere, and a cylinder. Each portion is heated in the oven to the same uniform temperature.

When he takes the portions out of the oven and leaves them at room temperature, which shape takes the longest time to cool down?

Details and Assumptions:

• Assume that the mashed potatoes are uniform in composition and that each shape is maintained throughout the process.
• The mashed potatoes are suspended in the air.

Paul has two identical ceiling fans, Fan A and Fan B, in separate but otherwise identical rooms. Fan A is switched on the entire time, while Fan B remains switched off.

After some months of use, which fan's blades will be dustier?


Assumptions: Upon sticking to the blade, dust is held by a strong electromagnetic force.

$$x$$ is a positive integer such that the sum of its digits times 5 equals itself.

What is $$x?$$

For example, 32 does not qualify because $$(3+2)\times 5 = 25 \neq 32.$$

Bonus: Prove that there is only one possible value for $$x.$$

Three hats are placed on the heads of three men standing in a line. Each man can only see the men and hats in front of him. In addition, the men all know that the hats came from a bag with 3 red hats and 2 blue hats.

Each man is asked if he knows which color hat he is wearing. The man in the back responds first, "No." Then the man in the middle responds, "No." Finally, the man in front responds, "Yes."

If they are all perfectly logical and telling the truth, what color hat is the man at the front wearing?

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