Problems of the Week

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2018-07-02 Basic

         

An artist has a bird-shaped gallery. To ensure the museum is fully guarded, he places 4 guards at the corners with red dots so that every inner wall is visible by at least one guard.

Can he fully guard the museum with fewer guards?

Seven identical regular hexagons are arranged in a honeycomb pattern.

If each hexagon has an area of \(8,\) then what is the area of \(\triangle ABC?\)

You need 8 liters of petrol. Using two containers that can hold 12 and 9 liters, you can do any of the following actions as many times as you like:

  • fill a container to its brim;
  • completely empty a container;
  • pour petrol from one container to another.

Is it possible to get exactly 8 liters (in either container)?

Two identical rows of \(\num{1000}\) dominoes are initiated in two different ways:

  • In row A, the first domino is barely tipped over, and hits the second domino in \(\SI{1}{\second}.\)
  • In row B, the first domino is pushed so that it hits the second domino in \(\SI{0.3\bar{3}}{\second}.\)

How long will the two rows take to finish falling?

This picture does not necessarily show optimal play.

This picture does not necessarily show optimal play.

My friend and I play a game in which we take turns placing identical coins on a circular table one at a time.

  • I place the first coin on the empty table.
  • The coins are placed flat on the table. They cannot overlap or go over the edge of the table.
  • The person who places the last coin on the table wins.

Which player can always win if they play optimally?

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