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## 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.

## Other useful links

- https://www.cheenta.com/gaps-in-permutation-tomato-objective-problem/
- https://www.youtube.com/watch?v=DR__BfDxNmc