# Finding the minimum number of boxes to pack products

• Chao Li
In summary, the problem asks for the number of boxes needed to pack a given number of items into three different sized boxes.
Chao Li

## Homework Statement

This was a coding challenge, and I've already completed it. But I feel there must be a better solution to my approach at the moment. Especially maybe there's a mathematical approach.

You have 3 sizes of boxes, small, medium, large:
small: holds 3 items
medium: holds 5 items
large: holds 9 items

If some one wants N items, how many x small, y medium and z large boxes to you need to pack that exact number, while ensuring that you use the minimum number of boxes?

I've solved this using loops, but I'm just wondering if anyone have a mathematical solution for this?

## Homework Equations

3x + 5y + 9z = N
solve for x, y, z such that x + y + z is a minimum

## The Attempt at a Solution

C:
int xlim = number/smallPackSize;
int ylim = number/mediumPackSize;
int zlim = number/largePackSize;

// loop thru all possibility
for (int x = 0; x <= xlim; x++) {
for (int y = 0; y <= ylim; y++) {
for (int z = 0; z <= zlim; z++) {
int totalProduct = smallPackSize*x + mediumPackSize*y + largePackSize*z;

// if total product number equals what we requested, save this combination.
if (totalProduct == number && totalProduct != 0) {
smallPackNo.add(x);
mediumPackNo.add(y);
largePackNo.add(z);

totalPackNo.add(x + y + z);
}
}
}
}
<Moderator's note: edited to add code tags. Please use them when posting code.>
Then just find the x,y,z combination that is a minimum from the lists.

This solution works. But I am concerned that it will be slow with large numbers. Does anyone have a more efficient method?

I think for a large number ##N## one can say use as much large sized packages as possible. So maybe it is faster to

1) calculate the number of large packages needed (## N/9 = ... ##) and skip the digits after the dot
2) calculate the residual items (## N\;mod\;9 = x ##), where ##x## only can be a number beyween 1 and 8 (if it is 0, the task is solve)
3) and then find rules depending on ##x## (e.g. if ##x = 8## then use on additional medium sized and one additional small sized package or if ##x = 1##, then decrease the number of large sized packages by one and add two medium sized packages instead, ...)

## 1. What is the purpose of finding the minimum number of boxes to pack products?

The purpose of finding the minimum number of boxes to pack products is to optimize the use of space and resources, ultimately reducing costs and increasing efficiency in the packaging process.

## 2. How is the minimum number of boxes calculated?

The minimum number of boxes is calculated by considering the dimensions of the products and the size of the boxes available. It involves finding the most efficient arrangement of products within the boxes to minimize empty space.

## 3. What factors should be taken into account when determining the minimum number of boxes?

Factors that should be considered include the dimensions and weight of the products, the size and shape of the boxes, and any specific packaging requirements or restrictions.

## 4. Can software or algorithms be used to find the minimum number of boxes?

Yes, there are various software programs and algorithms that can be used to calculate the minimum number of boxes needed for packing products. These tools can save time and provide more accurate results compared to manual calculations.

## 5. How can finding the minimum number of boxes benefit a company?

Finding the minimum number of boxes can benefit a company by reducing packaging costs, optimizing storage space, and improving the overall efficiency of the packaging process. It can also lead to a more sustainable and eco-friendly approach to packaging.

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