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
Hide Hint Show Hint Hide Answer Show Answer
What Next?
View a Similar Brain Teaser...
If you become a registered user you can vote on this brain teaser, keep track of which ones you have seen, and even make your own.
Solve a Puzzle
Comments
How can you have 27 equally-sized cubes? You can't.
Well, in that case, I AM THE ALPHA
... and my comment was interrupted, but here it continues:
---- Out of one cube you can make 27 equal cubes of one-third the size of the big cube, by 6 plane cuts (of 0 thickness). You get three layers of 9 cubes each (here represented by O):
OOO
OOO
OOO
OOO
OOO
OOO
OOO
OOO
OOO
---- Out of one cube you can make 27 equal cubes of one-third the size of the big cube, by 6 plane cuts (of 0 thickness). You get three layers of 9 cubes each (here represented by O):
OOO
OOO
OOO
OOO
OOO
OOO
OOO
OOO
OOO
Omega u r dumb. U can obviously cut a cube Z^3 number of simillar cubes where Z is a positive real number. Neway i quite like problem, would be slightly more interesting i think if you just asked people to find for what value of N the two options are equally expensive but thats a bit harder i suppose.
To dice a cube into smaller cubes you need to make the same number of cuts on each of the three axis, parallel to two faces for that axis. Each cut of a cube (or any parallelepiped) that is parallel to a pair of faces, creates two additional surfaces equal to the size of the parallel faces.
If the surface area of the original cube (or parallelepiped) is A, then each set of three cuts creates new surfaces with an area of A and the surface area is expressed as A*(n+1) where n is the number of cuts on each axis. Since the number of solids is (n+1)^3 the surface area will be A*(n+1)^(1/3).
This is a long way of explaining that for n divisions of the solid, the surface area is the cube root of n x the surface area of the solid. This holds as long as the faces of all the solids are parallelograms.
If the surface area of the original cube (or parallelepiped) is A, then each set of three cuts creates new surfaces with an area of A and the surface area is expressed as A*(n+1) where n is the number of cuts on each axis. Since the number of solids is (n+1)^3 the surface area will be A*(n+1)^(1/3).
This is a long way of explaining that for n divisions of the solid, the surface area is the cube root of n x the surface area of the solid. This holds as long as the faces of all the solids are parallelograms.
seems pretty straight forward, but I did have to break out pencil and paper. Thanks for the fun teaser.
I just ran into this again, and my previous comment doesn't simplify the problem in the manner I prefer.
Each time you cut the cube along all three axis, you create new surface area equal to the original cube. Therefore you will have 5 times the original surface area after you have made 5 cuts along each axis. This yields 5^3 = 125 small cubes.
Each time you cut the cube along all three axis, you create new surface area equal to the original cube. Therefore you will have 5 times the original surface area after you have made 5 cuts along each axis. This yields 5^3 = 125 small cubes.
To post a comment, please create an account and sign in.
Follow Braingle!