### Brain Teasers

# Little Cubes

You have 2 identical cubes. You cut the first cube into smaller, equally sized cubes and leave the second cube as it is.

You have two options: to paint all the small cubes with silver OR the remaining big cube with gold. Gold paint is five times more expensive than silver paint.

Which of the two options is cheaper, if the first cube is cut into

a) 27 b) 125 c) 1000 small cubes?

You have two options: to paint all the small cubes with silver OR the remaining big cube with gold. Gold paint is five times more expensive than silver paint.

Which of the two options is cheaper, if the first cube is cut into

a) 27 b) 125 c) 1000 small cubes?

### Hint

To get the amount of paint needed calculate the surface areas of the small cubes and the big cube.If the length of the big cube's side is x, its surface area is 6 * x^2.

If a big cube is cut into 3x3x3 = 27 small cubes, the length of a small cube's side is x/3.

### Answer

a) Paint the small cubes.b) Choose whichever option you wish, they're both equally expensive.

c) Paint the big cube.

If you have N small cubes and a big cube's side length is X, the side length of a small cube is X divided by the cube root of N.

The big cube's surface area is

6X^2

The small cubes' total surface area is N*6*(X/cbrt(N))^2

So if you have 27 small cubes, their total surface area is

27*6*(X/cbrt(27))^2

= 162*(X^2/3^2)

= 162X^2 / 9

= 18X^2

Multiply the big cube's surface area with 5 (gold paint is 5 times more expensive than silver)

Big cube "price" compared to the small cubes' price

a) 30X^2 > 18X^2

b) 30X^2 = 30X^2

c) 30X^2 < 60X^2

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