Any Brainiacs Out There? Math Question.

[flobadob]

If I have a large box, and I am going to fill it with smaller boxes, is there any way to work out the best way to pack the smaller boxes so as to maximise the space in the larger box and find out how to fit the most amount of small boxes possible.

For reference the large box has internal dimensions of 40cm Height, 38cm Width and 58cm Length. The smaller boxes have external dimensions of 12cm Height, 12cm width and 16cm Length.

At the moment the best I can do is fit 39 of the smaller boxes into the larger box and still be able to close the lid.

But is there a formula out there that will allow me to safely squeeze in a couple more??

[Peter Pedant]

oh god I used the word algorithm
fifty AB heads wont explode like in a scene from Scanners will it?

[ToraToraTora]

PP: "I thought someone must have written an algorithm for this " - this is well known as a very difficult algorithm in computing.

[ZebuSanctuary]

>06.54 // there's no simple formula //

There would appear to be some exceptions.

Given the l,w,h measurements of the smaller boxes fitting exactly and respectively into the l,w,h dimensions of the larger cuboid,
then no unused space within the cuboid would present itself.

Therefore the optimum number of smaller boxes packaged could be easily obtained as demonstrated by >14:19

Similarly, a smaller box in the shape of a cube allows the optimum number to be found straight forwardly.

Note, where the larger cuboid (l,w,h) dimensions were not a multiple of the side of the cube, rounding down would be the order of the day.

Clearly the orientation of the smaller boxes is a factor, this engenders different unknowns in unused space, couple this
with a miscellany of boxes of different sizes being put into the container along with their dimensions in rational numbers,
it is highly unlikely a polynomial algorithm exists to determine the optimum. Since computer science was alluded to earlier,
they will appreciate problems of this nature lend themselves to 'NP Completeness'.

The link below illustrates a program that is probably based on a heuristic or approximation algorithm.

https://www.engineeringtoolbox.com/smaller-rectangles-within-larger-rectangle-d_2111.html

Markedly, this program computes (very quickly) a number of smaller rectangles (of one type) which fit into a larger rectangular container, however not the optimum number.

Enter the following dimensions into the fields provided;

58 (width) and 40 (height) - Large Rectangle and

16 (width) and 12 (height) - smaller rectangle.

Space between rectangles = 0

It does not take much imagination to envisage if the dimensions for depth (38 and 12) are added, the container only caters for a total of 30 boxes. In relation to OP question this is clearly not the optimum.

JJ109 demonstrated 32 boxes. With a little rearrangement, my effort yielded 33 boxes - See link;

https://ibb.co/f4Xzc5N

Hope this helps.

[flobadob]

Hi everyone, very sorry for not keeping up with this thread.

I measured the large box on the day, but it's now obvious to me that the width has some wiggle room because I am able to squeeze 3 boxes into the width. Two of them are sideway and the 3rd is lengthway.

Because both boxes are cardboard I've been able to get an extra 2cm.

But I think the consensus is that either way I'm at the limit getting 39 into it.

Thanks for all the contributions.