Math Is Fun Forum

peterbill

Member

Registered: 2019-04-22

Posts: 1

Find the greatest number of boxes which can be packed in the crate

Can anyone help me- I don't understand why the answer is 124 and not 125

Question: A crate is a cube with side 1 m. A box is a cuboid which is 40cm by 20cm by 10cm. Find the greatest number of boxes which can be packed in the crate.

So the volume of the crate must be 1m^3. And the cuboid volume must be 0.4 m x 0.2 m x 0.1 m = 0.008m^3.

to find out how many boxes fit in the crate I thought it would be: 1 / 0.008 = 125

however the answer is 124- why is this!?

any help would be highly appreciated

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Find the greatest number of boxes which can be packed in the crate

hi peterbill

If the boxes were liquid, so that they could occupy any volume, regardless of shape, then you'd be correct.  But the boxes are rigid so it's not just a case of comparing volumes; you must also find a packing arrangement that works.  Two of the dimensions (20 and 10) divide into 100cm but the third (40cm) doesn't.  So you cannot just pack all the boxes the same way round without overflowing the space.

I started by filling a space 80 x 100 x 100 with 100 boxes and then tried fitting more by placing them the other way round. I only got to 120 and the remaining space doesn't have a 40 cm length to fit any more.  I'm thinking that a more complicated packing arrangement is needed but I haven't found it yet.  And I don't think there's a standard procedure for finding one.  I think it's just a case of trial and improvement.

LATER EDIT: Here's a way of packing 124:

There's a 20x20x20 space that I cannot fill.  That doesn't prove that 125 is impossible.  Not sure if that proof can be formed.  Thinking ……………...

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Find the greatest number of boxes which can be packed in the crate

OK, I have a proof. And a simpler packing arrangement.

I'll introduce some 'notation' to make it clearer.

With the crate in front of our position so we're facing one side, I'll call the left-right direction x, the back-front direction y, and the up-down direction z.

And to indicate which way round a box should be placed, I'll give its dimensions x first, then y then z.

Start by placing boxes 40x20x10.  You can make two entire walls of boxes like this taking up 80cm of the x direction.  This packs 50 x 2 boxes.

But no more can be placed with 40 in the x direction because 100 / 40 goes twice with a remainder of 20cm.

So now place boxes 20x40x10.  This fills across the front and we can make two columns of these packing another 10 x 2 boxes.  Then we hit the remainder problem again … there's a 20cm gap at the back in the y direction.  So no more boxes can be placed so that the 40 goes in the y direction.

Only choice left is to pack them with the 40 in the z direction: 10x20x40.  You can get another 2x2 like this and then there's only a 20cm gap left so no more can be placed with the 40 in the z direction.  That's all three directions filled to the maximum extent so maximum number of boxes is 50 + 50 + 10 + 10 + 2 + 2  = 124.

QED.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob