General / Off-Topic Puzzles

glynie

glynie
So the idea is that if you know any good puzzles, riddles, or anything worth giving a go, post them here and people can have fun trying to work through them. Obviously since it's my idea I should start off with one, so here's a little puzzle I made (more of a maths question really) that I found most people struggled with. See how you fare:

m2WOtMB4VMDLl_gHHQoM-1g.jpg


Imagine a large cube made of smaller cubes measuring 5 across each dimension (pictured above). The question is, if you were to pick any two smaller cubes at random, what are the odds that they would be adjacent (any of their faces would be touching one another)? Include the obscured interior when considering your answer.
 
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Rampant

Rampant
I don't know whether weffriddles is still active online. I really enjoyed going through those puzzles. Took me a loong time...

Cheerz

Mark H
 

Andy Kappa

Andy Kappa
Just for the sake of clarity: are adjacent cubes allowed to be different sizes (which would allow, for example, 48 1x1 cubes to be simultaneously adjacent to the same 4x4 cube)? Also, could you give a more precise definition of 'adjacent' please - are partial crossovers allowed? Finally, are we speaking strictly about cubes or are cuboids allowed?
 

tawittle

tawittle
5x5x5 = 125 cubes.
Calculating the number of "outside" cubes:
On each of the 6 faces, there are 9 cubes showing only 1 face. There are 12 edges total, each made up of 3 minicubes that show 2 faces each. Then there are 8 corners, each showing 3 faces. That means there are 27 inside cubes.

Any of the minicubes on the "inside" would have 6 cubes to pick. The face cubes showing 1 face have 5 cubes to pick, the edges have 4, and corners have 3.

Doing the calcuation:

(27/125)*(6/125)+(54/125)*(5/125)+(36/125)*(4/125)+(8/125)*(3/125)= .0384

Which is a little bit less than 5/125.

Do I win? :)
 

Andy Kappa

Andy Kappa
...and while I'm here, does the term 'smaller cubes' mean 1x1x1 cubes? Or does it mean any cubes of side less than 5? (There's lots of ways you can have 2x2x2, 3x3x3 and 4x4x4 cubes.)
 
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Andy Kappa

Andy Kappa
Really?

I'm very familiar with this as a 2-dimensional puzzle, and actually used to teach something similar to school Maths students. I'm not at all sure we can consider it a safe assumption. In fact part of the skill from the student was to see beyond first impressions and work out what could be done with it. Indeed that was my first thought as I looked at it. Makes it a more difficult puzzle of course...

ETA: For the record, I do understand your thinking. Maybe it's just that the wording is a bit lax from a mathematical standpoint.
 
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Zieman

Zieman
Well, to me it looks like the wording implies those 1x1x1 cubes the large cube is made of.

But as you say, there is ambiquity, and it may be a trick question in that regard.
 

Andy Kappa

Andy Kappa
Well, assuming they are 1x1x1 cubes, I get the following results:

Total number of possible pairings (excluding duplications) = (124x125)/2 = 7,750

Total number of possible adjacent pairings = 64x3 + 48x2 + 12x1 = 300

Chances of adjacent pairing = 300 out of 7750 = 6 out of 155 = 0.0387 (4 d.p.)


Not thoroughly checked so I may have missed something, but it feels right.
 
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glynie

glynie
Andy Kappa said:
Well, assuming they are 1x1x1 cubes, I get the following results:

Total number of possible pairings (excluding duplications) = (124x125)/2 = 7,750

Total number of possible adjacent pairings = 64x3 + 48x2 + 12x1 = 300

Chances of adjacent pairing = 300 out of 7750 = 6 out of 155 = 0.0387 (4 d.p.)


Not thoroughly checked so I may have missed something, but it feels right.
Click to expand...
That's the one, well done.

Sorry to anyone who was confused, I thought I explained it pretty well. Just a 5x5x5 cube made of smaller cubes. I didn't know how else to describe it. Adjacent means next to each other and I also said with faces touching so...I didn't know how better to describe that either. Anyway, well done Andy Kappa.

Basically it's all about considering the amount of faces each cube has exposed to another. So for example, there are 8 corner cubes and they only have 3 sides exposed, and the 2nd time you pick a cube there's one less to pick from, so that would be (8/125 * 3/124). When you add all of these together it looks like this:

(8/125 * 3/124) +
(36/125 * 4/124) +
(54/125 * 5/124) +
(27/125 * 6/124)

which gives as pointed out above, 6/155.

I remember having some arguments about this question actually. People didn't believe I had the right answer so, doubting myself, I made a simulator to check it. 6/155 is definitely correct.
 
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Andy Kappa

Andy Kappa
I actually got there by a slightly different route, but hey!

As to the query, the best way to make it explicit is to identify the smaller cubes as 1x1x1. Looks a bit tedious, but it instantly removes any doubt.

There used to be a range of investigative tasks that were used in schools to get students thinking about properties of complex patterns that weren't immediately obvious and had to be kind of hunted down. Often used as part of a larger investigation into number series'.

Example: Imagine a 5x5 grid drawn on paper. How many squares are there?

More pertinent example: how many cubes in a Rubik's cube? (Assume a cubical internal structure and not the mechanism you find in an actual Rubik's cube.)
 
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tawittle

tawittle
D'oh. I had all the probabilities for picking the second cube over 125 instead of 124, leading to my answer of .0384 instead of .0387.

And thats why you don't do puzzles at 4 a.m. kids :)
 

tawittle

tawittle
Anybody want to give this puzzle a shot?

Suppose you have every possibility for a shape of area 5 made up of 5 1x1 squares. (That makes 12 unique shapes altogether). The shapes are pictured below.

PentominoesPieces.jpg

The question is: if you have to use each piece exactly once, what dimension rectangles can you construct from the 12 pieces? Using a shape multiple times, omitting a shape, or having "gaps" in your big rectangle are not allowed.
 

glynie

glynie
Andy Kappa said:
I actually got there by a slightly different route, but hey!

As to the query, the best way to make it explicit is to identify the smaller cubes as 1x1x1. Looks a bit tedious, but it instantly removes any doubt.

There used to be a range of investigative tasks that were used in schools to get students thinking about properties of complex patterns that weren't immediately obvious and had to be kind of hunted down. Often used as part of a larger investigation into number series'.

Example: Imagine a 5x5 grid drawn on paper. How many squares are there?

More pertinent example: how many cubes in a Rubik's cube? (Assume a cubical internal structure and not the mechanism you find in an actual Rubik's cube.)
Click to expand...
Aye your method was interesting. Good point regarding the 1x1x1 description too.

Bazziegamer said:
What gets wetter as it dries?
Click to expand...
Still thinking. :)

EDIT: It's not like...
...dry white wine or something is it?
ANOTHER EDIT:
It must be a towel, or a sponge. :)
tawittle said:
Anybody want to give this puzzle a shot?

Suppose you have every possibility for a shape of area 5 made up of 5 1x1 squares. (That makes 12 unique shapes altogether). The shapes are pictured below.

Image

The question is: if you have to use each piece exactly once, what dimension rectangles can you construct from the 12 pieces? Using a shape multiple times, omitting a shape, or having "gaps" in your big rectangle are not allowed.
Click to expand...
Well...5x12 = 60, and the factors of 60 are 1,2,3,4,5,6,10,12,15,20,30 and 60...so I'm thinking only 3x30, 4x15, 5x12 and 6x10? (As all numbers have to be => 3 to fit the shapes in.)
 
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