Assume you have a huge cube. All the 6 sides of the cube are painted red. Then 6 straight cuts are made across the cube to give 125 cubes of equal size.
a) find the number of cubes with 3 sides painted red.
b) find the number of cubes with 2 sides painted red.
c) find the number of cubes with 1 side painted red.
d) find the number of cubes with no side painted red.
e) If all the cubes are put into a bag, and one cube is removed at random, what is the probability that its surface pointing towards you is red in colour?pls explain your answers
A maths puzzzle? explain me?
a)...well the only sides that can have 3 sides painted are corners, so that would be 8
(The rest, i think i might be misinterpreting the question...6 straight cuts are made across the cube...on each side? I don't get it. to get 125 cubes I imagine that each side of the cube has 5 cubes across and 5 down? though that would make for 4 cuts on each side so i don't know...)
if we go with 5 cubes lenght, width, height (to give 125)...then i say the following...
b) only edges can have 2 sides painted. so there are 5 cubes on each edge minus the 2 corner pieces (because they have 3 sides painted) and you have 5-2=3 cubes for each side. there are 12 sides to a cube. 12 x 3 =36
c) one side painted red would be the 'leftovers' of each outer surface. each outer side has 25 smaller cubes. 16 of them are edge pieces however (4 corner pieces, 12 edge pieces) and have more than 1 red side. 25-16=9, so there are 9 one side painted pieces for each of the 6 surfaces of the larger cube. 9 x 6 = 54
d) no sides painted would be the leftovers of everything else. so 8+36+54=98... 125-98=27 there are 27 pieces with none
e)
Reply:mmm the right answer has been given many times, so let's make it more fun: suppose we cut the cube into n x n x n smaller cubes. (In your case, n = 5 :) )
The answers are now
a) 8
b) 12 x (n - 2) = 12 n - 24
c) 6 x (n - 2)^2 = 6 n^2 - 24 n + 24
d) (n - 2)^3 = n^3 - 6 n^2 + 12 n - 8
e) we could add together
3 x 8 + 2 x (12 n - 24) + 1 x (6 n^2 - 24 n + 24)
but is faster simply to realize that there are 6 n^2 squares.
The probability is therefore
6 n^2 / 6 n^3 = 1/n.
Reply:how the heck do u make 6 cuts and get 125 cubes?
i'm making the following assuming you make 3 cuts each way
resulting in each side being cut into a 4x4 grid
thus 64 cubes
a) the eight corner pieces
b) the remaining edge pieces. There are 12 edges, each of them have 2 cubes (that are not corner pieces) thus 24
c) the remaing face pieces. 6 faces * 4 cubes per face = 24
d) the eight inside pieces
e) P(get a 3 side piece)*P(get red on 0 side piece)
+P(get a 2 side piece)*P(get red on 2 side piece)
P(get a 1 side piece)*P(get red on 1 side piece)
P(get a 0 side piece)*P(get red on 0 side piece) =
8/64 * 3/6 + 12/64 * 2/6 + 12/64 * 1/6 + 8/64 * 0/6 =
.0625 + .0625 + 0.03125 + 0 = 0.15625 = 5/32
...ok I gues the general consensus is that you are making EIGHT cuts for 125 cubes: thus
a) the eight corner pieces
b) the remaining edge pieces. There are 12 edges, each of them have 3 cubes (that are not corner pieces) thus 36
c) the remaing face pieces. 6 faces * 9 cubes per face = 54
d) the 3x3x3 = 27 inside pieces
e) P(get a 3 side piece)*P(get red on 0 side piece)
+P(get a 2 side piece)*P(get red on 2 side piece)
P(get a 1 side piece)*P(get red on 1 side piece)
P(get a 0 side piece)*P(get red on 0 side piece) =
8/125 * 3/6 + 36/125 * 2/6 + 54/125 * 1/6 + 27/125 * 0/6 =
0.032 + 0.096 + 0.072 + 0 = 0.2 = 1/5
I guess this is the same answer as Heather and Michael M got.
Reply:I think you should have changed the number of slices .. however I will go with 5X5X5 in the end.
a) 8
b) 36
c) 54
d) 27
e) There are a total of (6)5^2 = 150 painted sides. There are a total of (6)(5^3) total sides. (6)5^2 / (6)5^3 = 1/5 = 20%
Reply:the answer to all is NULL
you cannot get 125 cubes from a single cube with only 6 straight cuts.
125 cubes are derived from a cube that divided with 4 cuts along each of the x, y, and z axes, for a total of 12 cuts.
ok - that was my smartypants answer. You are patient...
6 cuts gives you 27 cubes.
a) 8 have 3 red sides (the corners)
b) 12 have 2 red sides (the center of each edge)
c) 6 have 1 red side (the center of each face)
d) 1 cube has no red (the very center)
162 total cube-surfaces, 30 of them are red. Any one of them has an equal chance of being drawn pointing towards you. 30:162 reduces to 1:5.4
Let's say you meant 12 cuts, not 6, for a 5x5x5 cube.
a) 8 have 3 red sides (the corners)
b) 36 have 2 red sides (the centers of the edges, but there are 3 cubes between each 3-sided corner, this time)
c) 54 have 1 red side (the center of each face is a 3X3 grid, repeated 6 times)
d) 27 cubes has no red (the center of the cube is a 3X3X3 block of unpainted cubes)
750 surfaces (125 X 6), 150 of them are painted red, any one of them has an equal chance of being drawn face up. 150:750 resolves to 1:5 - better odds than with the 3X3X3 cube.
Reply:A) As stated only the corners would have 3 sides painted, 8.
B) There would be 36 cubes with 2 sides painted.
C) There would 54 cubes with 1 side painted.
D) There would be 27 cubes with no sides painted.
E) The probability of pulling a cube out with the surface pointed towards you painted would be: 1 in 5.
If you imagine that the cube you're starting with as being 5 cubic units (Length, Width, and Height or X,Y,Z) it might help. Then make your slices. Each edge would consist of 2 (each painted on 3 faces) corner cubes that are shared by two other edges, between each of the corner cubes you would then have 3 cubes (each painted on 2 sides).
A cube has 6 faces, 8 corners, and 12 edges. Hence; each face of the original cube has a 3x3 square of cubes with one face painted (9 cubes/face * 6 faces), the original cube has 8 corner cubes with 3 faces painted, and each edge has 3 cubes between 2 corner cubes with 2 faces painted, which gives you 36.
54 + 36 + 8 = 98
125 - 98 = 27 cubes with no faces painted.
(54 * 1)+(36 * 2)+(8*3) = 150 total painted faces
125 * 6 = 750 total faces (painted and unpainted)
750/150 = the odds of selecting a cube with a painted face towards you.
Which you can reduce to 5/1, which then gives you a 1 in 5 chance of pulling a block with a painted face toward you.
Reply:for it to be 125 cubes it would have to be a 5*5 cube
a) each corner has 3 sides red, so 8
b) only the sides of the cube have 2 sides painted red, so 3+3+3+3+3+3+3+3+3+3+3+3=36
c) the 9 middle ones on each side, so 9*6=54
d) the rest of the cubes, so 125-8-36-54=27
e) 1/6 chance of seeing red for a one-sided cube, 1/3 chance of seeing red for a 2-sided cube, 1/2 chance of seeing red for a 3-sided cube. 0 chance of seeing red for a 0-sided cube. 8/125 chance of getting a 1-sided cube, 36/125 chance of getting a 2-sided cube, 54/125 chance of getting a 3-sided cube, 27/125 chance of getting a 0-sided cube. (1/6*8/125)+(1/3*36/125)+(1/2 * 54/125) +(0*27/125). i dont have a calculator but you can just type that in
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