How many boxes can fit in a van

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munro raymaker

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So this is from my sons math exam. The question is: You are moving away from home and have rented a van. The van has the internal dimensions:
Length 260 cm
Width 155 cm
Height 185 cm.
How many moving boxes can you fit inside the van when each moving box measures 40x40x70 cm? The boxes can be stacked in any way possible.

a. 36
b. 48
c. 54

My son said a, but his classmate said 48. My guess is 54. Who's right?
 
If one of the choices is correct, then one of you is right.
Show us your work and we can tell you if you are correct or where you went wrong.
 
We don't know the correct answer. It was a written exam for all the 9th graders.

My way of getting 54: To maximize the height I place one box lengthwise and two boxes standing up. That is 40+70+70=180cm. To maximise width I place two boxes lengthwise and one box sideways. That gives me 40+40+70=150cm. I continue filling up the van this way until there's no more space. This leaves a small gap at the end where i can place 4 boxes sideways stacked. In total 54. See image below.
But if we calculate the volume of the van and divide with the volume of each box there can be 66 boxes if the boxes were "fluid".

8D428845-B009-40BF-A303-E4CD162F09A3.png
 
munro raymaker said:
We don't know the correct answer. It was a written exam for all the 9th graders.

My way of getting 54: To maximize the height I place one box lengthwise and two boxes standing up. That is 40+70+70=180cm. To maximise width I place two boxes lengthwise and one box sideways. That gives me 40+40+70=150cm. I continue filling up the van this way until there's no more space. This leaves a small gap at the end where i can place 4 boxes sideways stacked. In total 54. See image below.
But if we calculate the volume of the van and divide with the volume of each box there can be 66 boxes if the boxes were "fluid".

View attachment 26953
Click to expand...
Just to check, in the 260x155x185 (LWH) van you have the one pile of 26 (brown, purple, red) taking up most of a (70*3)x(40*2)x(70*2+40) = 210x80x180 space, with 5 cm headroom; that leaves 155-2*40 = 75 in width, so you make a pile of 24 (blue) that is (40*6)x(70)x(40*4) = 240x70x160, with 25 cm headroom, leaving only 20 cm at the end and 5 cm in the width. Then you add a pile of 4 (green) to fill the 260-210 = 50 cm at the end, 40x70x(40*4), with 25 cm headroom again. Everything fits, and you've used 26 + 24 + 4 = 54 boxes.

I can't picture expecting students to do all that work on an exam, trying all ways to arrange the boxes. (That would be sadistic, since you can never be sure you can't do better, and there is no formulaic method.) I had spent some time looking at mixed arrangements not as elaborate as yours, but the most I could find was 48. I'd expect that to be the answer they want -- except that yours does work, so 54 must be the intended answer if it isn't just a matter of chance. (Or maybe there's a way to get even more!)

How much time did they have to solve this?
 
Well they had 3 hours in total to solve 7 long tasks in areas of arithmetic, trigonometry, statistics and equations. This was just one of them. You're right the most obvious answer would be 48. This question seems more like an intelligence puzzle. This exam was set for danish students by the way.
 
Packing problems are hard. I see no reason to assign this to ninth graders. The answer of 48 is not obvious, and your answer of 54 is ingenious. But this kind of problem, that very few students can solve correctly, just turns kids off math.
 
munro raymaker said:
Well they had 3 hours in total to solve 7 long tasks in areas of arithmetic, trigonometry, statistics and equations. This was just one of them. You're right the most obvious answer would be 48. This question seems more like an intelligence puzzle. This exam was set for danish students by the way.
Click to expand...
Actually, to my mind, it is not a test of intelligence but of perseverance. I'm guessing it is not an exam for a course, but a contest or an admission exam for some elite program for students who are willing to do hard things. In that setting it might make sense.

And since, as I said, there is no definite end point to the work (as any number less than 66 seems feasible, and it is impossible to be sure you haven't missed anything), even perseverance is not enough. It would also require the maturity to balance an attempt at perfection in this problem against giving the time to others. (Admittedly, as a multiple-choice problem, which in general I despise because they are not realistic, once you get 54 you can stop -- if you trust the author of the problem and are not obsessive.)
 
You could remove the top 3 green boxes (leaving one at the base) and replace them with 4 vertical boxes. Then you've got option (d) 55 !

Of course, all the options are actually correct because the question doesn't ask for the maximum number of boxes that could fit in the van, it just asks, "how many can you fit". :p :sneaky:
 
Cubist said:
You could remove the top 3 green boxes (leaving one at the base) and replace them with 4 vertical boxes. Then you've got option (d) 55 !

Of course, all the options are actually correct because the question doesn't ask for the maximum number of boxes that could fit in the van, it just asks, "how many can you fit". :p :sneaky:
Click to expand...
Brilliant!
 
Here's another bid from a clever work collegue of mine.

500.

Nothing is said that the boxes can't be folded. If we fold a box to a flat square of 110x110cm and we assume a box is about 1cm high folded, that means we can stack two horizontal piles of 185 boxes. Then fill the side with two vertical stacks of 155-110=45. Then add 40 boxes at the end because we have a gap of 260-110-110=40.

185x2+45x2+40=500.

Of course nothing gets moved then except the boxes?
 
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