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4 puzzles

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1) A line is drawn diagonally across every (individual) side of a cube, from one corner to another corner. How many patterns are possible, taking into consideration every possible orientation of the diagonals, including all six sides of the cube in every pattern?

post-48597-0-46007800-1317688687.gif

2) A tetrahedral blob of clay is sliced by six PERFECTLY straight (planar) cuts, the pieces not moving from their original positions. What is the maximum number of tetrahedral pieces that can be formed, counting only the pieces not further subdivided?

post-48597-0-45058800-1317688660.gif

3) Hollow victory is to Pyrrhic as hollow village is to [x]?

4) Universe is to cosmo- as universal laws is to [x]?

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1. 64 not counting symmetries

2. Two, because it's the only way "not further subdivided"

3. Empty

4. Omni-

Unfortunately, those are still incorrect.

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Why?

I'll begin with the analogies. "Omni-" is a prefix that means "all." It does not mean "law." A "pyrrhic victory" is one in which the side of the victor sustained such heavy losses that it can hardly qualify as a victory. Without giving too much away, the answer to "hollow village" is a name that is related to it in the same fashion that "hollow victory" is to Pyrrhic. As far as the cube calculation goes, it's simply not correct. For the tetrahedral problem, consider the definition of "subdivided" and place it in the context of "further."

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Let's take the cube calculation. There are two possible diagonals for each face. There are 6 faces. So, ignoring symmetries, there are 26 possibilities for drawing diagonals. If you are asking more than this you should state it unambiguously.

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Let's take the cube calculation. There are two possible diagonals for each face. There are 6 faces. So, ignoring symmetries, there are 26 possibilities for drawing diagonals. If you are asking more than this you should state it unambiguously.

Why did you ignore symmetries? Its difficulty is predicated upon its ambiguous phraseology.

Edited by addedline8

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Why did you ignore symmetries? Its difficulty is predicated upon its ambiguous phraseology.

I ignored symmetries as you had not stated that they are not to be ignored. Ambiguous phrasing is not the rule in mathematics and logic, which is what this forum is about. Clarity is important unless loosened parameters are stated up front. By the way, why did you ever thing that I thought "Omni-" meant "law"?

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I ignored symmetries as you had not stated that they are not to be ignored. Ambiguous phrasing is not the rule in mathematics and logic, which is what this forum is about. Clarity is important unless loosened parameters are stated up front. By the way, why did you ever thing that I thought "Omni-" meant "law"?

I disagree with your statement that ambiguity in mathematics/ logic is bad, problems with parameters that are not clearly defined leaves more room for analytic thinking and creativity. I assumed you thought "omni-" meant "law" because that is how you answered the analogy whereby you supply a prefix that is to "universal laws" as "cosmo-" is to universe.

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i got up to

62..though i think there are more but through the (1,2)(1,3).... method i got to 62

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I think

1)for the cube 32 patterns ignoring symmetry

2)4 regular tetrahedrons.

Depending on where the cuts are made they can be of same size.

Otherwise 1 big/small and other three of same size.

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Its 12 i guess for the tetrahedron

two groups of irregular tetrahedrons

6 in group 1 similar

other 6 similar in group 2...

{earlier got confused with pyramid and tetrahedron :P}

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Someone else posted the same tetrahedral problem after you, I also answered it there. The maximum number is

10 tetrahedrals, with four octahedrals thrown in the mix. To do it, you need to notice that the tetrahedral has 4 sides, but 6 edges. You must cut of roughly triangular prism shaped chunks on every edge, approximately 1/3 of the way through the tetrahedral. I don't have an image of this, but there is a way to describe it.

Instead of 6, assume we have 8 planar cuts. We make two of these cuts parallel to each side of the tetrahedral, again, each layer being separated by 1/3 the width of the tetrahedral. This forms three layers from top to bottom. On the top sits a 1/3 scale tetrahedral. Underneath that is a octahedral with three tetrahedrals attached to it. (The top part of the tetrahedral thus mentioned lookes like you just cut off each of the four corners of the original tetrahedral). And finally, the bottom has three octahedrals with 6 tetrahedrals located on the 3 corners and 3 sides.

while I used 8 cuts to describe how it looks, it actually can be done as I described with 6 cuts (1st paragraph)

I really need to find an image of this. >(

Edited by Palustrius

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3) Hollow victory is to Pyrrhic as hollow village is to [x]?

Spoiler

or  potyomkin (if the Russian pronunctiation is considered)

4) Universe is to cosmo- as universal laws is to [x]?

Spoiler

cosmonomo-

 

Edited by rocdocmac

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