# Cubes in a box

### Cubes in a box

I'm programming a simple app where I have to calculate how many cubes can fit in a box. I'm trying to figure out the formula to calculate how many cubes can fit in a box given width and height properties.
The cubes are all same size (width and height).

I have tried (bw * bh) / (cw * ch) where b is Box and c is Cube to calculate the areas, but in case where bw = 6, bh = 1 and cw = 2, ch = 3 you'll get (6 * 1) / (2 * 3) = 1 but in reality such cube can't be stored in this box.

So, for example if Box is 2x2 and cubes are 1x1, I could fit 4 cubes, but for 2x6 and 4x4 I would get 0, even though they have same area.

Thanks
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### Re: Cubes in a box

cubes, as well as your box, have three dimensions. You talk about "width and height" but what about length?
Guest

### Re: Cubes in a box

The formula you're using to calculate how many cubes can fit in a box is based on the ratio of areas, but it doesn't take into account the dimensions of the box and the cubes in a way that ensures they can be physically arranged inside the box. To accurately calculate how many cubes can fit inside a box, you need to consider both the area and the dimensions of the box and the cubes.

Here's a more accurate approach to calculate the maximum number of cubes that can fit inside a box:

Calculate the area of the box (BoxArea) by multiplying its width (bw) and height (bh).

Calculate the area of a single cube (CubeArea) by multiplying its width (cw) and height (ch).

Divide the BoxArea by the CubeArea to find the maximum number of cubes that can fit in the box.

However, to ensure that you're not overestimating the number of cubes that can fit, you also need to consider how the cubes are oriented within the box. For example, if the cubes are taller (height greater than width) and the box is wider (width greater than height), the cubes may not fit horizontally. Therefore, you should consider both the horizontal and vertical orientations of the cubes within the box.

Here's a modified formula that takes both dimensions and areas into account:

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MaxCubes = min(floor(bw / cw) * floor(bh / ch), floor(bw / ch) * floor(bh / cw))
This formula calculates the maximum number of cubes that can fit by considering both horizontal and vertical orientations and choosing the smaller of the two possibilities.

For example:

If the box is 2x2 and the cubes are 1x1, you would get MaxCubes = min(2 * 2, 2 * 2) = 4, which is correct.
If the box is 2x6 and the cubes are 4x4, you would get MaxCubes = min(0, 0) = 0, indicating that no cubes can fit in this configuration.
This formula should give you a more accurate result for calculating how many cubes can fit inside a box considering their dimensions and orientations.

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### Re: Cubes in a box

To calculate how many cubes can fit in a box, you need to consider both the width and height of the box and the cubes. If the width and height of the box are both divisible by the width and height of the cubes, then you can fit a certain number of cubes in the box.

The correct formula would be:

Number of cubes=⌊Box Width/Cube Width⌋×⌊Box Height/Cube Height⌋

This formula takes the floor of the division of the box width by the cube width, and the floor of the division of the box height by the cube height, and then multiplies them together. This ensures that only whole cubes are counted, without any partial cubes being included.

For example:

If the box is 2×2 and the cubes are 1×1, then 2×2=4 cubes can fit.
If the box is 2×6 and the cubes are 4×4, then 2×6=12 cubes can fit.

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