2018-04-16 Basic

         

With pipe 2 closed, pipe 1 will empty the water tank in 1 hour.

With pipe 1 closed, pipe 2 will empty the water tank in 2 hours.

If both pipes are open, how long will it take for the water tank to empty?

Is it possible to form 3 identical squares with four matchsticks of length 1 and two of length \(\frac12?\)

Note: You are allowed to cross the matchsticks, but you are not allowed to break them.

Dr. Frankenthree creates numbers by splicing three consecutive positive integers together. He notices these numbers are multiples of 3:

\[ \begin{align} {\color{blue}7}{\color{red}8}{\color{green}9} &= 3 \times 263 \\ {\color{blue}14}{\color{red}15}{\color{green}16} &= 3 \times 47172 \\ {\color{blue}50} {\color{red}51}{\color{green}52} &= 3 \times 168384 \\ {\color{blue}198} {\color{red}199}{\color{green}200} &= 3 \times 66066400. \end{align}\]

Are the numbers Dr. Frankenthree creates in this way always multiples of 3?

I was once on a Canadian lakeshore in late autumn, and the frozen water appeared perfectly clear.

I haven't seen a mountain lake freeze since, and I've never been successful at making clear ice cubes at home, so I've decided that the environmental conditions of that autumn must've been just right.

What local conditions are necessary to form clear ice on a lake?

This question is part of the Brilliant.org Open Problems Group where we try to solve problems nobody knows the answer to. You can join in the discussion here.

Multiple copies of a polyomino can sometimes be used to form a rectangle.

Using multiple copies, which of the polyominoes below can be formed into a rectangle?

Note: You can only use copies of the designated type, and while rotation and reflection are allowed, overlapping is not.

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