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This is the problem I came across reading the book The Art and Craft of Problem Solving. When I read this question I wasn't able to figure out the solution and I saw the solution after a while, but still I couldn't understand the solution as well.

Question:

Pat wants to take a $1.5$-meter-long sword onto a train, but the conductor won't allow it as carry-on luggage. And the baggage person won't take any item which greatest dimension exceeds $1$ meter. What should Pat do?

Solution:

This is unsolvable if we limit ourselves to two-dimensional space. Once liberated from Flatland, we get a nice solution : The sword fits into a $1 \times 1 \times 1$ -meter-box, with a long diagonal of $\sqrt{(1^2 + 1^2 + 1^2)}$ = $\sqrt{3} > 1.69$ meters.

Can anyone give me a clear explanation of this solution?

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  • $\begingroup$ Lol 19 seconds and three answers... you got more than what you asked for probably $\endgroup$ – skv Oct 18 '14 at 16:14
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    $\begingroup$ Use it to kill the conductor. That should do it. $\endgroup$ – ralian Oct 22 '14 at 22:30
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The sword fits in that 3D box, because the long diagonal is $1.69$ meter. The longest dimension of this box is 1 meter (length, width and height are all exactly 1 meter). The sword goes like the green line in the picture:

Box

If Pat would use a flat case, it would be denied. In the most naive way, the box would be $1.5$ meter long and have a width of the sword itself. A more clever way to think is in the maximum dimensions, which are $1 \times 1$ meter. In that case the diagonal is $\sqrt{2} = 1.41$ meter long, which is not long enough. So the only option is to include the height into account.

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    $\begingroup$ The greatest dimension of a cube is it's diagonal so technicality it should be refused too $\endgroup$ – Seb Oct 23 '14 at 16:40
  • $\begingroup$ Technically the dimensions are the height, width and depth. These are all 1 meter long. The trick is that inside these dimensions you can have longer lengths. $\endgroup$ – Mathias711 Oct 23 '14 at 16:59
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    $\begingroup$ Incorrect, the most commonly cited dimensions are height, width and depth, sure, but for the purposes of 'it's maximum dimension', the diagonal is counted $\endgroup$ – Jon Story Oct 31 '14 at 17:14
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    $\begingroup$ @JonStory No, dimensions are in no way related to the longest line element in the volume (or whatever thing) spannend by the unit vectors. The definition 'dimension' that is used here is mathematical, not as a length. $\endgroup$ – Mathias711 Oct 31 '14 at 17:34
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Explanation

The solution given hinges on the fact that, while the sword itself is longer than the maximum allowed length, we can construct a volume in which the sword fits, that doesn't exceed the restrictions. Contradictory, we can work around the restrictions that were undoubtedly enforced to save space, by using more space!

I've actually run into similar limitations once when sending a parcel and I solved it in the exact same way.

Escalation

But a $1\text{m} \times 1\text{m} \times 1\text{m}$ box is a bit overkill. The diagonal is almost $20\text{cm}$ longer than needed!
So let's find a couple of smaller solutions.

Minimal height

Let's say we have a box with a base of $1\text{m} \times 1\text{m}$. What is the minimal height $h$ we need?

The length $L$ of the sword is $1.5 \text{m}$. The diagonal $D = \sqrt{1^2 + 1^2 + h^2}$. If we fit the sword precisely in the diagonal, $L=D$, so $1.5=\sqrt{1^2 + 1^2 + h^2}$. That means that $2.25 = 2 + h^2$ and so $h=0.5\text{m}$.

Minimal square

If we want a box of exactly $1\text{m}$ high with a square base, how long would the edges $x$ of that square be?

This means that $D = \sqrt{1^2 + x^2 + x^2}$, so $\tfrac{9}{4} = 1 + 2x^2$. This makes $x^2=\tfrac{5}{8}$ and $x=\tfrac{1}{4}\sqrt{10} \approx 0.79\text{m}$.

Minimal cube

Let's try to get the minimal cube with edges $x$ in which to fit the sword.

In this case, $D = \sqrt{x^2 + x^2 + x^2}$, so $\tfrac{3}{2} = \sqrt{3x^2}$, which means that $\tfrac{9}{12} = x^2$, which yields $x = \tfrac{1}{2}\sqrt{3} \approx 0.87\text{m}$.

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  • $\begingroup$ What’s missing: minimum volume box. May well be one of the ones described but a proof would be nice $\endgroup$ – theonetruepath Feb 23 '18 at 5:04
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The baggage person won't allow the sword, because it's 1.5m long. They will, however, allow a 1m x 1m x 1m box. If you put the sword running from one corner of the box to the opposite, it will fit (because the diagonal is ~1.7m) and be allowed on board.

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