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What is the maximum number of rectangular blocks, each with dimensions 12 centimeters by 6 centimeters by 4 centimeters, that will fi t inside rectangular Box X ? The inside dimensions of Box X are 60 centimeters by 30 centimeters by 20 centimeters.

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What have you tried so far? –  Hans Parshall Apr 25 '11 at 16:31
    
@Hans If the box is resting on a side that is 30 cm by 20 cm, then 30/6,20/4 × = 5 × 5 = 25 blocks will fi t on the bottom layer. In this case, the height of the box is 60 cm and 60/12 = 5 layers will fi t inside the box. –  prem shekhar Apr 25 '11 at 16:45
    
@prem shekhar: This is what I've suggested in my answer below, so I guess I don't see where you're having trouble. Let me repeat from below that if you've completely filled the box, then you've used the maximum number of blocks! –  Hans Parshall Apr 25 '11 at 16:46
    
@Hans If the box is resting on a side that is 60 cm by 30 cm, then 60/12 , 30/6 × = 5 × 5 = 25 blocks will fi t on the bottom layer. In this case, the height of the box is 20 cm and 20/4 = 5 layers will fi t inside the box. –  prem shekhar Apr 25 '11 at 16:49
    
@Hans ya ya..in this way in any way we will get 25*5=125 –  prem shekhar Apr 25 '11 at 16:50

1 Answer 1

up vote 1 down vote accepted

Draw a picture!

Pick a bottom for the box, and try to fill that bottom completely. If you can stack the pattern that fills the bottom completely a number of times that evenly divides the height, you've completely filled the box. If you have no leftover space, you've certainly used the maximum number of blocks.

Hint: You can fill the box in your problem completely.

Here's a similar example. Say I had $1 \times 2 \times 3$ blocks and a $2 \times 4 \times 6$ box. Then along the bottom ($2\times4$) of the box, I need to decide if I want the bottom of the blocks to be $1\times2$, $2\times3$, or $1\times3$. I can fill the bottom of the box with 4 blocks by choosing $1\times2$ bottoms. By stacking this pattern twice, I've filled the entire box!

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