I was wrapping some presents earlier, and the problem of optimal wrapping is actually less trivial than it appears. I'd imagine that a fairly large proportion of presents are cuboidal, and that most people would place the present so that its sides are parallel to the edge of the paper, and then wrap around it.
Wrapping a cube like this means that you need this much paper:
-------------------------------
| | | |
| X | | X |
| | | |
|---------|---------|---------|
| | | |
| | | |
| | | |
|---------|---------|---------|
| | | |
| | | |
| | | |
| X |---------| X |
| | | |
| | | |
| | | |
|---------|---------|---------|
The regions in the middle are the net of the cube, and the regions marked X are "wasted". Perhaps surprisingly, in this case the same area is wasted as the amount that is used!
People could perhaps cut out the net of the cube, and then use the corner pieces to wrap smaller presents, but I'd say it's rare that people do this.
I only realised this because I underestimated the length I would need to wrap a present, failed to cover it the "normal" way, and started considering alternatives.
As Warwick Dumas says, wrapping diagonally is indeed more efficient!
It makes for some interesting-looking presents though. For example, I have a (cuboidal) CD case wrapped in a present that is roughly a trapezoidal prism and the seams where the edges of the paper meet make an H shape instead of just a line.
]]>You've obviously put some work into it and you've seen fit to come onto the forum and defend it, so the least I can do is give you the benefit of the doubt; I know how math savvy the media can be.
]]>Re the area etc, the report that I wrote is publicly available on request. It should be understandable to anyone familiar with calculus and basic trigonometry, so I do encourage you to read it. The point of finding out the area is purely to compare the areas given by using different methods.
The person who said that the widthways remainder on the paper is important, is absolutely correct, and obviously this fact is not contained in a formula for the area used. The answer is that you should be wrapping two or more things alongside each other so that the remainder does not have to be wasted. (If you were not going to do this, wrapping diagonally would be superior.)
Thank you for your comments.
Warwick Dumas
]]>Now that I think about it... that wouldn't work. We already know the ideal width for the paper is A+C, meaning that the height of those flaps should be HALF of C.
So his equation should ACTUALLY be:
Area=2ab+2ac+2bc+(4)(0.5)(0.5c)c
A = 2(ab+ac+bc)+c²
regards
khushboo
]]>I would have loved to eavesdrop on the conversation when they asked how he arrived at this magic formula and (to cover for himself) he started going into how he used advanced differential calculus to find the ideal solution for three variables.
"Gimme my commission now, kthxbai!"
]]>Also, instead of longest to shortest, a to c meant length, width and height (meaning they were interchangable, and so they should be interchangable and so the formula not being symmetric doesn't make sense either).
At least your article makes mathematical sense, even if it's utterly pointless and not at all practically useful for the reasons you said.
]]>Formula created by University of Leicester researcher
...
Bluewater, the UKs leading shopping centre, discovered that Brits continually overestimate the amount of paper they need to wrap their Christmas presents. Following this new revelation, Bluewater today reveals the mathematical solution which will hopefully put an end to unnecessary paper wastage: A1 = 2(ab+ac+bc+c²)**
...
The formula has been created by Warwick Dumas from the Department of Mathematics, University of Leicester , who has been working with Bluewater to devise the perfect method of gift-wrapping that will help customers save time and money as well as reducing the amount of paper that will be wasted.
...
(**) A = area needed/ a, b, c = Dimensions of cuboid: a = longest, c = shortest
From: http://www2.le.ac.uk/ebulletin/news/press-releases/2000-2009/2007/12/nparticle.2007-12-04.6745557516
Right. Ok. The magic equation is Area=2ab+2ac+2bc+2cc. It took a university professor to tell us that the solution to how to wrap a cuboid is to use paper that has an area equal to the surface area of what we're trying to wrap (plus enough for some flaps on the end)?
I'm not sure why he threw in the 2cc? I know he put them in there for the little triangles on the end, but they won't be covering anything that wouldn't already covered, so this "solution" isn't really "[putting] an end to unnecessary paper wastage".
Now I suppose it's fair enough. Not all people are good at Math and not all people know how to work out the surface area of a cuboid, but even knowing this equation still isn't enough. We now need to find wrapping paper that is a+c (or 2b+2c) wide to actually use this equation or we end up either overwrapping or cutting off extra paper; in both cases there is wasted paper.
So... is this equation really fit for purpose? And how much do you reckon this professor got paid for telling Bluewater the equation for the surface area of a cuboid?
]]>