Given a 10x10x10 box what is the largest size you can make six boxes that will fit inside the original box.?

I have to interpret your statements, as you leave out info.

1. Assuming the 6 boxes fill the entire inside, or as much as can.

2. assuming the 6 boxes can be different

3. assuming "largest size" means the total volume of the 6 boxes

4. assuming there is no fixed shape for the boxes.

Then the answer is: the total volume of the boxes is a maximum of 1000 (10x10x10)

There is many different solutions. Here is one:

5 boxes of 10x10x1

1 box of 10x10x5

here is another:

4 boxes of 10x10x1

2 boxes of 10x10x3

.

The box sides can't intersect? Or the box volumes can't intersect?

The boxes can be different sizes, or must they be the same size as each other?

Do the boxes have to be cubes? Or can they just be... box shaped?

Answer 1: Assuming that the box edges mustn't intersect, but the volumes can, and that the boxes can be different sized cubes, and the thickness of the edges is less than 1.4, then I'll make 6 boxes:

9.75 x 9.75 x 9.75

9.5 x 9.5 x 9.5

9.25 x 9.25 x 9.25

9 x 9 x 9

8.75 x 8.75 x 8.75

8.5 x 8.5 x 8.5

and stack them inside each other.

Answer 2: If the boxes aren't allowed to stack inside each other, and don't have to be the same size, but they are all "box shaped" - i.e. they don't have to be cubes:

I make 4 boxes, each of size 5 x 10 x 3.25

and 2 boxes, each of size 5 x 10 x 3.5 and stack them together into the big box.

and stack them on each other.

That fills the big box.

Answer 3: If the boxes aren't allowed to stack inside each other, and do have to be the same size, and are all "box shaped" - i.e. they don't have to be cubes:

I make 6 boxes, each of size 5 x 10 x 3.25 and stack them together into the big box.

There's still left over 10x10x0.25 = 25 of space :-( can you do better?

Answer 4: If the boxes aren't allowed to stack inside each other, and do have to be the same size, and have to be cubes, then I make 6 boxes, each of size 5 x 5 x 5 and put them in 6 of the 8 corners of the big box. There is still left over 2 x 5 x 5 x 5 = 250 of space :-( can you do better?

Some assumptions:

1) largest size you can make 6 boxes means largest volume of EACH small box

2) the six boxes must fit exactly into the large box

3) increments of 1/4 means the dimensions of the small boxes can not be less than 1/4

The large box volume = 1000

Six boxes would each have a volume of 1000/6 = 166.67

thus each box would have dimensions 10x5x10/3 which gives a volume of 166.67

but 1/4 increment is not met, so do

4 boxes of dimension 10x5x3.25 with volume of 162.5 each, and

2 boxes of dimension 10x5x3.5 with volume of 175 each

Total volume of six boxes = 1000

39

I would make them like this:

9.75 x 9.75 x 9.75

9.5 x 9.5 x 9.5

9.25x 9.25 x 9.25

9 x 9 x 9

8.75 x 8.75 x 8.75

8.5 x 8.5 x 8.5