Cubes and Rectangles – Math Olympiad Hanoi 2018


Try this beautiful problem from Math Olympiad Hanoi, 2018 based on Cubes and Rectangles.

Find the number of rectangles can be formed by the vertices of a cube.

  • 6
  • 8
  • 18
  • 12

Key Concepts


Geometry

Permutation

Combination

Check the Answer


But try the problem first…

Answer: 12.

Source
Suggested Reading

Math Olympiad Hanoi 2018

Geometry Vol I to IV by Hall and Stevens

Try with Hints


First hint

There are 6 squares on 6 faces on the cube.

Second Hint

There are 4 diagonals of the cube that have the same length

and pass through the center of the cube. Every two diagonals intersect at the midpoint and form a rectangle.

Final Step

Then there are 6+ $(\frac{4!}{2!2!})$ =12 rectangles.

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