Puzzles with numbers and things

[Monique]

Anyone prospecting for gold dust?

Have been thinking of it but I'm not sure I understand the question actually.

[abextra]

Have been thinking of it but I'm not sure I understand the question actually.

Yeah... maybe we have another herring served...

[moglet]

I've been trying to think if Charlie could carve a ball that contained none at all or at least less than the required amount.

Do we have enough data? Herrings or not....

[abextra]

Well... I'm really bad at explaining... ... if Gordon injected all the dust 146 spects per injection, he had to make at least 343 injections... and then the distances between the injections would be so small that a snooker ball would contain more than one of them... maybe I misunderstood the question again?

[Monique]

according to the "story" all specks are within a cube of size 4x4x4 "ball diameter". So I calculated the "density" of specks per ball volume in that cube... 50100/64*(PI/6) and that's about 780 (I don't have the exact number here)
Now if for every possibe ball carved that ball contains less 147 specks ... meaning the speck density per ball is always < 147, I feel that the density in the inner cube can of course not be as high as 780 (or rather the exact number I found ...)

[moglet]

Does that follow though?, he could have planted them all close together in one corner (we are only told "at various points"), but very close together, Charlie would know this but somehow I don't think this is sufficient "proof"

[Monique]

No I'm not sure it's sufficient proof. And I'm not sure if D_G means that Charlie just knows it is possible to carve a ball or if he is actually able to carve it because he knows where to do that which is an entirely different question.

[abextra]

according to the "story" all specks are within a cube of size 4x4x4 "ball diameter". So I calculated the "density" of specks per ball volume in that cube... 50100/64*(PI/6) and that's about 780 (I don't have the exact number here)
Now if for every possibe ball carved that ball contains less 147 specks ... meaning the speck density per ball is always < 147, I feel that the density in the inner cube can of course not be as high as 780 (or rather the exact number I found ...)

I found the number is 782,81... but it would be the number of specks in a cube with side width of 1 ball, the volume of a ball is smaller and there would be a bit over 407 specks in a ball... if the specks were spread equally.

[Monique]

I found the number is 782,81... but it would be the number of specks in a cube with side width of 1 ball, the volume of a ball is smaller and there would be a bit over 407 specks in a ball... if the specks were spread equally.

Yes but even if they are not spread equally ... IF no ball - wherever carved - contains more than 146 ... average density in the cube can't be 782,81 because that would require an average of about 407 specks per ball

[moglet]

I guess we've all done the arithmetic, perhaps Charlie has a magic chisel

Are we seriously suggesting that Gordon has injected each speck individually? That would take more than the hours of darkness!!

[davis_greatest]

Have been thinking of it but I'm not sure I understand the question actually.

It sounds from the later post like you do. Gordon can inject the gold dust wherever he likes - you need to show that, wherever he injected it, as long as Charlies knows where that is, Charlie can find somewhere to carve the ball.

Yeah... maybe we have another herring served...

No herrings in this one

I've been trying to think if Charlie could carve a ball that contained none at all or at least less than the required amount.

That would depend on how Gordon distributed the dust. If he spread the specks evenly, then Charlie could not find a ball with fewer than 147 specks (or even quite a lot more than this). But if he put the whole lot near a corner, say, then Charlie could carve a ball with no specks. However, that is not the question here! :snooker:

Well... I'm really bad at explaining... ... if Gordon injected all the dust 146 spects per injection, he had to make at least 343 injections... and then the distances between the injections would be so small that a snooker ball would contain more than one of them... maybe I misunderstood the question again?

Getting warmish - at least with the numbers 343 and 146 - but Gordon could inject the dust anywhere - maybe only one speck per injection, maybe more, we don't know. (Only Charlie knows.)

according to the "story" all specks are within a cube of size 4x4x4 "ball diameter". So I calculated the "density" of specks per ball volume in that cube... 50100/64*(PI/6) and that's about 780 (I don't have the exact number here)
Now if for every possibe ball carved that ball contains less 147 specks ... meaning the speck density per ball is always < 147, I feel that the density in the inner cube can of course not be as high as 780 (or rather the exact number I found ...)

This is also warm. However, it's not good enough just to look at the density - Charlie needs to be able to carve a ball, so we need to look at the shape of the ball, not just its volume. (The marble isn't plasticine - it isn't malleable )

Does that follow though?, he could have planted them all close together in one corner (we are only told "at various points"), but very close together, Charlie would know this but somehow I don't think this is sufficient "proof"

If they were all in one corner, then it is clear that Charlie could carve the ball he needs. But you need to prove that Charlie can carve the ball no matter how the specks are distributed.


abextra's and Monique's answers combined are getting close to a proof. We know the specks must be in a 4x4x4 cube. Now, what is the largest cube that can fit inside a snooker ball? And then consider cubes slightly smaller than that....

[Monique]

I guess we've all done the arithmetic, perhaps Charlie has a magic chisel

Are we seriously suggesting that Gordon has injected each speck individually? That would take more than the hours of darkness!!

Not to mention that he managed to stay behind the cube unseen from Gordon while still seeing where he injected the specks ... he's seeing through the cube with magic eyes maybe.

[davis_greatest]

No I'm not sure it's sufficient proof. And I'm not sure if D_G means that Charlie just knows it is possible to carve a ball or if he is actually able to carve it because he knows where to do that which is an entirely different question.

Charlie saw exactly where each speck was placed (where it was injected and how deep, hence its location). So Charlie both knows that it is possible AND is able to carve it.

I found the number is 782,81... but it would be the number of specks in a cube with side width of 1 ball, the volume of a ball is smaller and there would be a bit over 407 specks in a ball... if the specks were spread equally.

Yes, there would be about 410 specks per ball IF the specks were spread evenly - but they are not necessarily.

I guess we've all done the arithmetic, perhaps Charlie has a magic chisel

Are we seriously suggesting that Gordon has injected each speck individually? That would take more than the hours of darkness!!

They might be injected individually or maybe he did them in batches. Either way, Charlie can carve the ball he needs. And Gordon is pretty quick when he needs to be!

[davis_greatest]

Not to mention that he managed to stay behind the cube unseen from Gordon while still seeing where he injected the specks ... he's seeing through the cube with magic eyes maybe.

He was wearing night vision goggles! I thought that was obvious.

[abextra]

Yes, there would be about 410 specks per ball IF the specks were spread evenly - but they are not necessarily.

But if they were NOT spread equally, then there were places in this 4x4x4 cube where there were more than 410 specks per ball?