Yuna finds a box with an open top. Each side is 8 inches long. If she fills this box with identical 2 inch by 2 inch cubes, how many of these cubes will be touching the box?

Answer 1

you have an 8x8x8 cube

there are 4 layers of 16 2x2x2 cubes

all cubes on the bottom layer touch the bottom of the box

on the second layer the 12 perimeter cubes all touch the box but the 4 center cubes do not

similarly for the third layer, the 4 center cubes don't touch the box

similarly for the fourth layer, the 4 center cubes don't touch the box

4+4+4=12 cubes don't touch the box

there are 64 cubes in the box

64-12=52 cubes touch the box

Answer 2

There are a total of 64 cubes [that is 4x4x4 cubes, for 2" cubes and 8" sides].

We can account for them as follows:

On the bottom layer, all 4x4=16 cubes touch the box. ---- 16 cubes

On the next three layers up, there are 12 cubes that form the perimeter of the layer -- 3*12 = 36 cubes. Since the box has no top, the top layer is counted just like the other two of these layers, touching only on the perimeter.

16 + 36 = 52 cubes touch the box

Now, just to check, the top 3 layers have 2x2 cubes inside the perimeter that DO NOT touch the box. 3*2*2 = 12 cubes



52 + 12 = 64 That's all of them

Answer 3

There will be five faces that have cubes touching the box (none will touch the top)

bottom face: four 2 inch cubes x four 2 inch cubes: 4 x 4 = 16 cubes

first "wall" 16 cubes less the 4 cubes already in place along the bottom: 16 - 4 = 12

second "wall" 16 cubes less the 4 cubes along the bottom and the 3 additional along the first wall:

16 - (4+3) = 9

third "wall" 16 cubes less the 4 cubes along the bottom and the 3 additional along the second wall:

16 - (4+3) = 9

fourth "wall" 16 cubes less the 4 cubes along the bottom, the 3 along the third wall, and the 3 along the first wall:

16 - (4+3+3) = 6

The total touching the five faces of the box is: 16 + 12 + 9 + 9 + 6 = 52

Answer 4

First, find the number of cubes touching the bottom: 3 rows of 3 each = 9.

Next, add one side---3 cubes are already there, so just add 6.

Next, add the opposite side---add another 6.

Finally, add the last two sides---2 each

Number of cubes touching the box = 9+6+6+2+2 = 25

Another way: There are a total of 27 cubes, and only 2 are not touching the box. If the box had a top, then only one would not be touching.