[bonanova]

You have a cube of cheese; each edge is a cm long.

You make a plane cut through the cube that creates a face having six sides.

That's cool, you think, let's see if I can do that again.

So you make a second plane cut, parallel to the first; it creates another face with six sides.

The cube is now in three pieces; one of them has two 6-sided faces.

How thick can that piece be?

[Guest]

an upper bound of a[sqrt(3)-2sqrt(2/3)], which is about .099a.

I'd need at least 3 diagrams to show how I got that, but am I on the right track?

[bonanova]

an upper bound of a[sqrt(3)-2sqrt(2/3)], which is about .099a.

I'd need at least 3 diagrams to show how I got that, but am I on the right track?

Sounds plausible. I've made a sketch but not done the math yet.

[Guest]

a[sqrt(3)-sqrt(2/3)] = 0.915 a

which looks a lot better (should be bigger than or around sqrt(3)/3)

Edited October 3, 2008 by bonanova
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[Guest]

You have a cube of cheese; each edge is a cm long.

You make a plane cut through the cube that creates a face having six sides.

That's cool, you think, let's see if I can do that again.

So you make a second plane cut, parallel to the first; it creates another face with six sides.

The cube is now in three pieces; one of them has two 6-sided faces.

How thick can that piece be?

The first cut would be across two corners, at a 45 degree angle off the x-axis. The next cut would be on the very opposite side, at the same angle (hence parallel). These two cuts could be infinitely close to the tips of the corners. So really, all you need to know is how far it is from one corner to the very opposite one. Which turns out to be 3 cm.

Am I right? Edited October 3, 2008 by bonanova
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[Guest]

a[sqrt(3)-sqrt(2/3)] = 0.915 a

which looks a lot better (should be bigger than or around sqrt(3)/3)

It looks like we did things very similarly, except

I had a[sqrt(3)-2sqrt(2/3)] = .099 a

The two comes from the fact that each cut needs to come in off the diagonal by that same amount - it's symmetrical.

Also, I used a spoiler

Edited October 3, 2008 by bonanova
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[k-man]

Hmmm...

a/sqrt(2). I will post a sketch when it's done

[Guest]

a face having six sides = a face with 6 edges ??? Like a hexagon ???

[k-man]

Here is my full answer. I just realized that I made a typo in my first answer Doh!

[spoiler=I like cuttin' cheese ]

Any plane lying between and parallel to the red and blue planes on the picture will cut the cube in the 6-sided shape. So the thickness of the 3-dimensional shape between these planes is the object of the problem. The line BG is perpendicular to the cutting planes and the points E and F are the points where it crosses these planes. Due to the symmetry we can only look at the pyramid ABCD and calculate BE (the height of the pyramid). Then EF will be equal to BG-2*BE and BG is known to be a * sqrt(3).

ACD is an equilateral triangle with the side equal to a * sqrt(2), so AC = AD = CD = a * sqrt (2)

From the equilateral triangle we can find that AE = DE = CE = a * sqrt(2) / sqrt(3)

Finally from the triangle AEB (angle AEB = 90 degrees) we can find that EB = a / sqrt(3)

Therefore, EF = a * sqrt(3) - 2 * a / sqrt(3). Simplifying we get EF = a / sqrt(3).

[Guest]

Here is my full answer. I just realized that I made a typo in my first answer Doh!

The OP said

You make a plane cut through the cube that creates a face having six sides.

I don't see ant face in your solution that has 6 sides !!!

[bonanova]

I don't see ant face in your solution that has 6 sides !!!

Aside from corners B and G, 3 of the corners are red and 3 are green.

Identify the 6 edges that connect a red corner with a green corner.

Mark the midpoints of these 6 edges.

The midpoints are coplanar, and you can slice the cube on that plane.

The face that cut exposes has six sides - a regular hexagon.

That face is perpendicular to BG, and it can "slide" along BG anywhere between the red and green planes.

That distance, which is the objective of this puzzle, is the largest thickness of a 6-sided section.

[k-man]

I don't see ant face in your solution that has 6 sides !!!

...of the cuts that create 6-sided faces.

Since the OP is looking to find the maximum thickness of the piece between the red and blue planes we need to get them as far as possible from each other. As you move them further away they become more and more like triangles with three of the six sides approaching zero length. In my post I showed the extremes and I said: "Any plane lying between and parallel to the red and blue planes on the picture will cut the cube in the 6-sided shape"

[Guest]

As far as I can see it should bejust less than half the length of diagonal of a side i.e. a/sqrt(2).

[bonanova]

Starbreaker has the approach,

k-man showed the derivation and has the right answer

CR and horia were close with the numbers [horia suggested it]

imran had a close guess.

Kudos to all.

The distance between k-man's red and green planes is exactly 1/3 of the cube diagonal.

To see this, stand a cube on a point so a cube diagonal is vertical.

Study the slopes of any three edges that connect the top and bottom points.

Symmetry arguments that become apparent as you study it say they all have the same slope.

They're also equal length - they are all cube edges of length a.

So the differences among the points vertically are the same = 1/3 of the cube diagonal.

The two interior points of the three edges lie one on the red and one on the green plane.

So the red and green planes are separated by a distance of 1/3 of the cube diagonal.

Now the "math" part.

If you don't know that the cube diagonal is a * sqrt[3], just apply pythagoras twice.

Max thickness = a * sqrt[3]/3 = a/sqrt[3].

[Guest]

Aside from corners B and G, 3 of the corners are red and 3 are green.

Identify the 6 edges that connect a red corner with a green corner.

Mark the midpoints of these 6 edges.

The midpoints are coplanar, and you can slice the cube on that plane.

The face that cut exposes has six sides - a regular hexagon.

That face is perpendicular to BG, and it can "slide" along BG anywhere between the red and green planes.

That distance, which is the objective of this puzzle, is the largest thickness of a 6-sided section.

Got it and thanks ... This by itself was a puzzle

[Prof. Templeton]

Got it and thanks ... This by itself was a puzzle

I agree. By the time I had visualized where the cuts had to be, several people had posted answers already.