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It's a grade 5 problem.

A box is 1.4m long, 1.2m wide and 90cm high. Small blocks of 20cm long, 12cm wide and 8cm high are place in the box. Find the maximum number of blocks the box can contain.

I'm not sure about the way to proceed to have the answer for the MAX amount of blocks.

N. F. Taussig
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Bheem
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2 Answers2

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As all the dimensions of the blocks are multiples of $4$cm, any axis aligned rectangular packing will have dimensions that are also a multiple of that. In particular, you will not be able to fill more than $88$ cm of the $90$ cm dimension. In that case, you can align the blocks so that $7$ of the $20$ cm dimension fill the $1.4$ m, $10$ of the $12$ cm dimension fill the $1.2$m, and $11$ of the $8$ cm fill the $88$ cm. $7 \cdot 10 \cdot 11=770$ blocks will fit.

We have not shown that there is an arrangement with the blocks not lined up with the axes that will pack more blocks. I am skeptical, but it is possible.

Ross Millikan
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$\frac{1.4\times1.2\times0.9}{0.2\times0.12\times0.08}=787.5$

So 787, right? No... The blocks are not liquid, so you need to find the setting that maximizes the amount of blocks leaving the least empty spaces.

Option 1: $7\times10$ blocks in each layer leaving no empty spaces and $11$ layers, thus $770$ blocks.

Rodrigo
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