Packing Silly Spray

Geometry Level 4

A company is trying to ship rectangular boxes of Silly Spray with square bases that have side length 3 and height ten inches. The larger boxes that they are shipped in are $23 \times 23 \times 23$ inches. What is the most amount of Silly Spray boxes that they can fit in the larger boxes?

Details and Assumptions

The smaller boxes cannot be at a tilt, they must be on their side or facing straight up.

The answer is 114.

1 solution

Robert Fritz
Feb 18, 2014

First, vizualize the size of the box compared to a can of silly string. Then think of how many cans can fit from each length of the box being 23. Since the diameter is 3, 3 goes into 23 the most 7 times. Then because the length of a can is ten. 10 goes into 23 twice at the most. Then notice that because there are 3 inches left, you can fit another 2 cans that are turned horizontally. Then because the height is 23 the cans stacked together are how many times 3 goes into 23 which is 7. Since one layer has 16 cans, the amount of cans per box would be 16 times the amount of layers which is 7. 16x7=112

You can fit at least 114 cans in each box.

In the bottom 10 inches, you can fit 49 cans.

https://i.imgur.com/Oz1OWLC.png

In the next layer of 10 inches, you can fit another 49 cans. Then in the top 3 inches, you can fit another 16 cans, as described in the solution above.

https://i.imgur.com/h1LmNrZ.png

This gives you a total of $49 + 49 + 16 = 114$ cans.

You can fit even more cans if for each 10 inch layer, you use a hexagonal packing instead of a square packing.

Jon Haussmann - 7 years, 3 months ago

Also perfect.

Finn Hulse - 7 years, 3 months ago

Which means I need to change the answer. :P

Finn Hulse - 7 years, 3 months ago

@Finn Hulse – If you want to update the answer, send me an email.

For packing questions like this, it can be tough to justify that you reached the maximum. The 2-D version is already tricky, and the 3-D allows for a lot of variations.

Calvin Lin Staff - 7 years, 3 months ago

@Calvin Lin – Yes, would you mind changing it to 114? Thanks. And I specified that you can't place them anyway except for horizontally and vertically.

Finn Hulse - 7 years, 3 months ago

@Finn Hulse – By horizontally I meant on its side.

Robert Fritz - 7 years, 3 months ago

I miss understand. i think that the cylinder cannot be placed horizontally. :P

Hafizh Ahsan Permana - 7 years, 3 months ago

@Hafizh Ahsan Permana – It can, because the length of the box is 23 in, and that means if you have 2 ten inch in length cans, there is now 3 inches left to which you can fit a can horizontally because the diameter of a can is 3 inches.

Robert Fritz - 7 years, 3 months ago

Perfect.

Finn Hulse - 7 years, 3 months ago

Packing Silly Spray

The answer is 114.

1 solution

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