A 10x10x10 cube is painted red on all faces and then cut into ten 10x10x1 slices. Each slice is then cut into twenty-five 2x2x1 blocks. how many of the 250 blocks have exactly one face painted red?

bruhlmao Dec 21, 2021

#1**+2 **

I think it's 104. If anyone gets a different answer feel free to tell @bruhlmao.

SmartWater Dec 21, 2021

#3**+2 **

Its not 104 let me try again

One eternity later

So if 10x10x10 is 10 times greater than 10x10x1 then it means the cube was cut into 10 equal pieces.

2 of the slices will have one of the big sides paint out because those 2 slices are on the end of the cube. Those 2 slices will have 5 sides painted unlike the remaining 8 in the middle that will only have 4. Now the area of each slice is 100 because 10x10x1 =100 and since each of the slices are cut into dimensions os 2x2x1 which equals 4.100/4 = 25 which means its area is 25.

10x10x1

2x2x1

From that comparison we can see that one dimension hadnt been touched. And it we look back at the begining if a cube is cut into 10 pieces like a loaf of bread then it should be like a square-like prism.and is we divide that its kind of like a cake. And it its like a cake the 1 in 10x10x1 could be the height of the cake. Some pieces will be from the middle and from the 8 slices we can see that is its kind of like a cake. Lets look at the 8 slices that are not on the edges of the separation. Its kind of like a cake with only the sides covered nothing else. And we know that each slice is sliced into 25 equal slices. and since the sides are covered the perimeting slices will have 1 side covered excluding the corners. and since we splitted the slice into 25 parts that means we have 5 by 5 to equal one of the 10 slices we split our cube into. each of those tiny slices should have a enge lenth of 2 by the way. and since 5+5+3+3 = 16 and we are excluding the corners because they will have 2 sides painted then its 16-4 = 12 cubes for each of the 8 slices. Which means 8 x 12 = 96 total slices for the 8 slices we split our cube into.

Now we are left with the 2 sides that have a extra side filled in. Which are one of the faces of the painted sube which means for each of the 2, 1 of the big faces are painted. So for these 2 extra slices, let's picture itas a cake with all of it painted exept the bottom. (Please picture the cake as a square prism for this explanation please) and now that the top is filled the perimeter of each of them the entire 16 piece perimeter has 2 or more sides painted. So lets look inside the perimeter. Since the top is covered and the bottom is not there should be a lot of 1 sided slices. and since it is everything exept the perimeter so if we count it we will get 9 and to confime that we know that 16 (The perimiting slices) Plus 9 (The non perimeting squares) equals all 25 slices. so now we know that for each of the 2 edging slices that have a face painted there are 9 for each of them. so 2x9 = 18

So now we have everything we need to solve this problem. We have the 8 slices and the 2 slices to make 10 total slices, the first slices we did to this fully painted cube. 96 (Total 1 sides tiny cubes for the 8 slices) and 18 (Total 1 sided slices for the 2 slices) Finally the last thing we have to do is add them, 18+96 = **114 as our answer**

I really hope this helped. I spent 35 minutes typing and solving this equation

Hopefully I can see you another time

SmartWater Dec 21, 2021

edited by
SmartWater
Dec 21, 2021

edited by SmartWater Dec 21, 2021

edited by SmartWater Dec 21, 2021

edited by SmartWater Dec 21, 2021

edited by SmartWater Dec 21, 2021

edited by SmartWater Dec 21, 2021

edited by SmartWater Dec 21, 2021