[Prof. Templeton]

Long before Professor Templeton started his career at Redrum University he had a job working in the warehouse of the Manhattan Fruit Exchange. One morning he arrived at work to find that an overhead oil line had begun to leak onto a stack of Kiwi boxes. The oil had leaked along the top of the stack and run down the sides and onto the floor where the uneven surface allowed it to run underneath as well. The boxed Kiwis were stacked into a cube (not necessarily of equal dimensions) and all the boxes on the outside of the cube were now stained with oil while all the boxes on the inside were fine. The Prof. was told to separate the damaged boxes from the undamaged boxes and get a count for insurance purposes. When the Prof. was done he noted that the separated piles were of equal quantities and less than 500 boxes remained undamaged. How many boxes were there to start with?

[Guest]

Before I go counting boxes, can you clarify something? What do you mean when you say the "cube (not necessarily of equal dimensions)"? I ask, because while I was working on bononova's cube puzzle over the weekend, I learned that a cube is "a regular solid having six square faces", or having all dimensions equal (there went my attempted solution...). So do you mean your cube may not be a cube but some other hexahedron, or do you mean the kiwi boxes themselves may not be cubes but when stacked together do form a cube whose dimensions may be formed from different numbers of non-cubic boxes?

Thanks!

edit: how did I manage to ask a question that is nearly as long as the OP?

Edited January 5, 2009 by Cherry Lane

[Guest]

moving along, and assuming you mean that the stack of boxes forms a hexahedron with non-square faces, I've found the answer to be

a stack of 12 by 10 by 8 boxes, with 480 damaged and 480 undamaged boxes.

I don't have a mathematical solution yet, but I'll get there.

[Guest]

moving along, and assuming you mean that the stack of boxes forms a hexahedron with non-square faces, I've found the answer to be

a stack of 12 by 10 by 8 boxes, with 480 damaged and 480 undamaged boxes.

I don't have a mathematical solution yet, but I'll get there.

The stack would have to be a hexahedron and not a cube otherwise there is no possible solution (in whole numbers) because...

x^3-6x^2+12x-8=(x-2)^3<500

which yields no solution in which x is a whole #

[Prof. Templeton]

Before I go counting boxes, can you clarify something? What do you mean when you say the "cube (not necessarily of equal dimensions)"? I ask, because while I was working on bononova's cube puzzle over the weekend, I learned that a cube is "a regular solid having six square faces", or having all dimensions equal (there went my attempted solution...). So do you mean your cube may not be a cube but some other hexahedron, or do you mean the kiwi boxes themselves may not be cubes but when stacked together do form a cube whose dimensions may be formed from different numbers of non-cubic boxes?

Thanks!

edit: how did I manage to ask a question that is nearly as long as the OP?

Six faces but not necessarily square or equal dimensions. So yes all the kiwi boxes are originally stacked into a hexahedron. The boxes themselves are also not necessarily cubes, but hexahedrons.

[Guest]

I found the following formula helpful for building a spreadsheet to home in on the solution. I just wasn't as speedy as some

: .

face cubes:

width * length * 2 + width * (height-2) * 2 + (width-2)*(height-2)= exposed face cubes

total = width * length * height

interior= total - face

[Prof. Templeton]

moving along, and assuming you mean that the stack of boxes forms a hexahedron with non-square faces, I've found the answer to be

a stack of 12 by 10 by 8 boxes, with 480 damaged and 480 undamaged boxes.

I don't have a mathematical solution yet, but I'll get there.

That's it. Your good at these, CL. There are 20 cuboids where the number of inside blocks equal the number of outside blocks, but only one under 1000 total blocks. Did you come across others when solving?