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Posts posted by md65536
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Then my previous answer is no good. I guess any pirate is satisfied if they get to divide the treasure, and any pirate is satisfied if they can pick, but obviously the others wouldn't be happy letting someone else pick.
SpoilerOne splits the treasure. Everyone else picks. The first picks from whatever wasn't picked. Any piles of loot that were picked by exactly one pirate are taken, and the remaining pirates repeat the process with the remaining loot. Each time, at least the pirate that divided the treasure is satisfied.
Edit: This doesn't work either, given your last post. I could divide them as 50%, 47%, 1%, 1%, 1%, hoping everyone would pick the 50% and I'd get 47%. Then the pirate who never gets to split won't be satisfied. Or someone will pick the the 47% and I won't be satisfied.
Edit2: I suppose they can make it work. The dividing pirate has to divide fairly if they want to be satisfied. Every other pirate, if they think the one who divided will end up with a pile that's more than a fifth, making subsequent rounds potentially unsatisfiable, they can just choose that one and be satisfied. All it would take is that they purposefully choose a satisfactory option rather than one that could cause dissatisfaction later.
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6 hours ago, Genady said:
If there were two pirates rather than five, a solution could be that one of them divides the treasure into two parts, and the other one chooses.
There are a lot of extra assumptions here. 1. the treasure can be divided to any precision. This wouldn't work if the treasure was 3 large diamonds. 2. The one dividing believes they can do so exactly. 3. There is no conflict in choosing who divides and/or who picks.
(I guess pirates are more peaceful than children.)
SpoilerA simple extension is just have one pirate divide, and let another pirate pick. Then repeat the process with the remaining 4 pirates.
This adds an assumption that there's no conflict in who divides and (but not or) who picks. I don't think this is a good solution, but without knowing what the actual assumptions are, I don't know what needs fixing. I don't think it would work in practice.
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4 hours ago, Genady said:
How long are they expected to stay in prison?
Spoiler2.4 years?
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7 hours ago, Genady said:
Yes, relativistic Doppler effect includes time dilation. As I know of a solution to the twin paradox that does not include relativistic Doppler effect, could you refer to a solution that does?
Sure, apologies for veering off-topic but the ideas still relate! If you look at the Doppler factor formula, you'll see why a sign change of v gives a reciprocal factor. Before or while looking at examples, here's a challenge. Using v=+/-0.6c, the Doppler factors are 2 and 0.5 (perhaps opposite of intuition?) and Lorentz factor is 1.25. Given a statement like, "B spends 1 year traveling away while seeing A age 0.5 years. B turns around, and spends 1 year returning during which it sees A age 2 years, for total of A aging 2.5 years to B's 2," can you similarly (with no more math than that) describe what inertial twin A sees? There's no need to calculate distance, delay of light etc.
https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html
https://hepweb.ucsd.edu/ph110b/110b_notes/node60.html
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On 5/21/2023 at 10:08 AM, Genady said:
Hmm ... Except the twin paradox has nothing to do with the Doppler effect. It has to do with the time dilation. The latter is the same regardless of direction of the motion. It depends only on the relative speed between two frames of reference.
The relativistic Doppler effect includes time dilation, and it gives a complete solution to the basic twin paradox (hard to hide aging when you can see each other age the entire time).
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4 hours ago, Genady said:
It reminded me of a puzzle with two racing bugs. One goes up and down a wall with constant speed. Another goes up, to the same height, with the half speed and then down with the double speed. Who wins?
The same thing happens with the relativistic Doppler effect in the twin paradox, with an outbound and return trip at the same speed. If the clocks appear to tick at 0.5x the rate of a local clock on the outbound trip, they'll appear to tick 2x on the inbound trip.
Someone once used incorrect intuition to argue on these forums that this shows that the clocks would age the same amount during the trip, thus disproving special relativity.
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12 hours ago, TheVat said:
At lower latitudes, the visualization is easier, once you know that in terms of apparent motion the sun is overtaking the moon. The moon's apparent motion is that it's going west across the sky more slowly since it is in fact orbiting Earth spinward, i.e. from west towards east. So its shadow also goes generally that direction since its motion relative to the sun's apparent motion is eastward.
I thought this might be incorrect reasoning, because you could change the east/west direction of the shadow simply by changing the rotation rate of the Earth, without affecting how the moon moves relative to the sun. But I think your reasoning must be right. It seems then that if the Earth were spinning much faster, then even though day-to-day the moon appears to have lagged behind the sun, during a single day the moon would appear to be overtaking the sun at lower latitudes. This would be due to parallax.
The animation at https://en.wikipedia.org/wiki/Solar_eclipse_of_August_23,_2044 shows how the shadow can go "backwards" at high latitudes. It seems that in this case, the sun and moon are "to the North", eg. during "night time" where there is midnight sun. The sun still appears to be overtaking the moon, but they're both moving in a west to east direction that late in the evening! This should happen anywhere at high latitudes when there is an eclipse in the evening after the sun has passed the westernmost point in the sky and begins moving eastward again before setting, or at dawn before the sun reaches the easternmost point.
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Spoiler
Given the hint, I figure the easy solution involves that at every moment, B is moving perpendicularly to A?
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You can ask ChatGPT about itself. It probably knows a bit more about it than scienceforums.net users. Parts of its answers:
QuoteAs an AI language model, I work by using a neural network to analyze large amounts of text and identify patterns and relationships between words and phrases. [...]
The training process involves feeding the neural network vast amounts of text data, such as books, articles, and websites, and then adjusting the weights and biases of the network based on the patterns and relationships it identifies in that data. This allows the network to gradually learn how to generate text that is similar to the text it has been trained on.
Much, but not all, of what it says is just based on what other people have said. For example if you ask it a science question that is generally misunderstood repeatedly across the internet, and is not something some higher-weighted "expert system" would describe correctly, you might likely get the wrong generally misunderstood answer.
Months ago I asked it if the Twin Paradox required general relativity, and it told me that it does, sounding just like a post you might see anywhere across the internet. Now if I ask it, it says it doesn't. I don't know if that's better data, better model, better weightings, or if it is actively being corrected on answers it commonly gets wrong. There is a small army of humans providing it with specific data, not just scraped stuff.
QuoteAs an AI language model, I have been programmed to perform a wide range of mathematical operations, including:
- Addition, subtraction, multiplication, and division of numbers.
- Solving equations and inequalities.
- Calculating percentages, ratios, and proportions.
- Finding the prime factorization of a number.
- Calculating logarithms and exponents.
- Solving systems of linear equations.
- Performing calculus operations such as differentiation and integration.
It is not as simple as taking everything that's been written and scraped, and generating something similar. It is programmed with a bunch of different specific abilities. For example, it can multiply 2 numbers no one has ever multiplied before, with no data in existence to suggest what the probable answer is based only on what's been written before. It could conceivably learn the steps to do something it's not programmed to, but in this case it has specific programming for it.
QuoteYes, as an AI language model, I was programmed with a special code specifically for generating poetry. This code is designed to help me identify and replicate the various structural and stylistic elements that are common in poetry.
The code for generating poetry involves using a neural network to analyze existing poems and identify patterns in their structure, such as the number of lines, the number of syllables per line, and the rhyme scheme. The neural network also identifies patterns in the language used in poetry, such as the use of metaphor, simile, and other figurative language.
Once the neural network has learned the patterns and structures of poetry, it can generate new poems by using those patterns and structures as a guide. The generated poems may not be identical to existing poems, but they will share many of the same features and will often have a similar tone, mood, and theme.
The general and most basic functionality is to mimic what it has seen before, and that alone can answer a lot of questions correctly and carry on a conversation similar to ones that have happened before. But there is a lot more additional programming that it hasn't just "learned" by itself. Certainly, additional programmed capabilities will be added over time.
An AI like this can create new ideas. For example, if someone somewhere has associated A with X, and someone else has associated B with X, it's possible for an AI to associate A with B, even if no human has ever done that.
When people say an AI doesn't, or never will, "understand" something like a human does, I wonder how they define understanding, or feeling, thinking, etc., without using "like a human does," in their definition. How do we know that human understanding is more than just the learned connections between a very large set of concepts and knowledge?
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I can't imagine ever noticing that, but now I see why it works for other triangles with the same areas. If you skew the triangle eg. to make a right-angle triangle, you don't change the areas. If you uniformly scale the triangle vertically by r (preserving the ratios between the areas), and then scale horizontally by 1/r, each region's area is scaled by the same factor.
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37 minutes ago, Genady said:
It doesn't happen to be so in general - it works here because we assume that the answer is completely determined by the given data.
SpoilerI think it does generalize, and that the answer is completely determined by the data because it generalizes. Or to put it another way, a+b = g+j for any inscribed triangle, regardless of the other data. The generalization is that if 2 intersecting lines are both tangent to a circle, the intersection point is equidistant to the 2 tangent points. I used that equality about 4 more times to solve it.
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Spoiler
7.8?
I made a right triangle that fits the specification, but it got pretty complicated. I haven't figured out how to use the clue!
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That makes it a lot easier! Is it
Spoiler28?
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I see there's other ways to figure this out, but I noticed that there are lots of ways the DE line can be chosen...
SpoilerAnd if you maximize the length DC, EC goes to 0 and you get a degenerate triangle with perimeter 2 DC. But you can also maximize EC and get perimeter 2 EC. But if any DE line works, those maximums would have to be the same length... does that always happen in general?
Anyway the answer I get is
Spoiler8
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20 hours ago, Genady said:
instead of removing points from a 1x1 square we find where to put a point.
SpoilerI feel like I'm missing something simpler than this:
Say you have any shape of area < 1. Make a new shape by cutting the entire plane into 1x1 tiles with cuts along whole values of x and y, then overlapping all tiles and their pieces of the original shape, with a corner at (0, 0). Since the total area < 1, there must be some point inside that 1x1 tile that isn't covered by the new shape, and if you move both shapes together (without rotation) so that such a point is on a grid vertex, then neither shape is touching the grid.
Intuition is that rotation will complicate things because it allows for more ways in which a shape can be moved to dodge the grid, but it seems not to matter because it can still be placed off the grid without rotation.
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48 minutes ago, Genady said:
But I have already hinted that the latter is correct.
No fair asking for something that's impossible in the original problem 😠
"Proof":
SpoilerAssume there's a shape with area <1 that touches any 1x1 grid, placed on the plane. Place down all possible grids of the same orientation, so that the plane is completely covered. Consider the square from (0,0) to (1,1), with size 1. For all grids, if that grid touches the shape, remove all points on the grid, also removing the point from the square.
When done, if you haven't removed all the grids then the assumption was wrong, and there is a grid that doesn't touch the shape. If you've removed all the grids, you've removed at least one point from the shape for every point in the square you removed, so you removed at least an area of 1 from the shape, and the shape must have had an area of at least 1.
I think there might be a problem here talking about the area of points, like with the Banach-Tarski paradox.
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On 3/7/2023 at 7:15 AM, Genady said:
Find a shape with area < 1 such that it cannot be placed on the plane without touching at least one of these points.
SpoilerI'm thinking of a very thin very long and very slight curve that gradually increases in curvature, so that for a given length L, every possible length n*L for some integer n, where n*L is less than the length of the shape ... ??? can be found on the shape no matter how it's oriented. I'm not sure if that's the right criteria. Then if it works for all L in [1, sqrt(2)], it should touch at least one point. Is this on the right track? I feel like I got it backwards, and I'm missing something involving the width of the shape.
We're trying to prove 2 opposite things. Does trying to prove that an arbitrary shape can be placed without touching a point, lead to a contradiction?
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How did you solve it?
I noticed a pattern in what each subsequent term was "leaving out", and figured out a formula for the sum of the first n terms, then used induction to see that it works as n goes to infinity. Is there another way?
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6 hours ago, Genady said:
The whole point of the analysis is that you need to consider not only the length contraction effect, but relativity of simultaneity.
I concur. This sort of thing has cropped up before, and it has always been due to relativity of simultaneity (usually).
Forget about the fluid for now. Consider the frame where both Vs are traveling toward each other at the same speed. They're length-contracted to an identical shape, and should therefore collide at all points along the V simultaneously. Therefore neither is there an enclosed volume to trap fluid nor extra space for the fluid to escape.
The same events happen in all frames, ie. all points along the Vs must collide. The main difference is that the events aren't all simultaneous in other frames. Yes, in one frame the point will contact first, and in another the ends collide first.
When you add the fluid back, in all frames the outcome is the same: either the fluid is moved early enough so it can escape in all frames, or it doesn't have enough time to move in any frame.
You can maintain a paradox by stipulating some impossible requirement. Eg. have the 2 Vs come to mutual rest instantly in multiple frames. Eg. accelerate a perfectly rigid body (including a volume of incompressible liquid).
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11 hours ago, Markus Hanke said:
You couldn’t describe the geon itself using Newtonian gravity - only its effects on external test particles that are sufficiently distant.
I think that applies to all elementary particles in Newtonian gravity.
16 hours ago, md65536 said:it must be that when separate bits of energy combine to make up a rest mass, the density distribution of that mass is not necessarily the same as the density distribution of the energy that makes up it.
I think I've been tricked into showing that I was wrong all along. Even with a Newtonian approximation of GR, adding energy somewhere does not necessarily simply add a Newtonian gravitational effect there. Even when the added energy adds rest mass to a system, it might not behave as if the mass was added where the energy was.
Back to the original problem of 2 masses, it's possible it works approximately well when the masses pass, ie. with the system treated as a particle that gains rest mass as measured from a distance. However, when adding energy to each of the 2 masses, the behavior is not the same as if you increased their individual rest masses.
To try to get away from the Newtonian view, mass is not some kind of "stuff" that exists independently, it's just the measurement.
11 hours ago, Markus Hanke said:when I answer people’s GR questions on this forum, I am very careful about giving intuition-based predictions about what happens in specific scenarios; in fact I try to avoid doing so altogether, unless I have the mathematical abilities to actually check the prediction
Of course I'd rather have simple answers but I suppose that's best, especially when bad intuitive explanations can stick around for decades, being repeated in pop sci media etc.
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On 3/16/2023 at 5:42 AM, Markus Hanke said:
How about an interior metric, such as FLRW [...]
Definitely a Newtonian approximation has an error amount and it seems completely useless in some of these cases.
An example I was thinking of is if you have 3 equal masses in an equilateral triangle say arranged in a v shape, and a test mass balanced in the center. Then you increase the masses of all 3 with the mass at the bottom very slightly more than the others. With Newtonian gravity the test mass would be expected to accelerate downward. But suppose that the increases of mass on the top 2 were from rotation, with each spinning opposite of the other so that the test mass would instead fall directly upward due to frame dragging.
That might work fine or not??? because as explained in this thread the effects are non-linear. You can't just add up or overlay 2 separate frame-dragging effects. So there may be some additional energy in the interaction of the 2 spinning masses (it might even be that to get the test mass to move upward, it still needs greater mass increase "upward"???), or some other thing I can't imagine. I guess either "there is an answer given just the information above, but it's more complicated than suggested" or "the answer changes depending on factors other than just which of the 3 masses is given more rotation or energy." I have no idea which.
On 3/16/2023 at 5:42 AM, Markus Hanke said:Or how about something like a gravitational geon - a topological construct that is held together purely by gravitational self-energy, without the presence of any other energy-momentum sources at all? This particular solution relies entirely on GR self-interaction effects - under Newton, a completely empty spacetime without any gravitational sources cannot contain (or maintain) gravity.
This is an example of how we're talking about different things.
Mass in Newtonian gravity is based on how it's measured, such as how a mass fits gravity equations. It's not based on a prediction of what Newton's laws say it should weigh. For example we wouldn't say "The Newtonian mass of an atom is a lot less than the measured mass of an atom because it doesn't include the contribution of binding energy." Geons have mass, and Newtonian gravity can approximate its effect.
Suppose there's some elementary particle that has a measured mass, and behaves approximately as Newtonian gravity predicts. Say, for argument only, along the lines of Wheeler's speculation, that it is discovered the particle is actually a geon. We wouldn't say, "According to Newtonian gravity, this particle can now be considered massless and is predicted not to gravitate, contrary to measurements."
Mass in Newtonian gravity necessarily includes contributions of things that are only explained by GR. Meanwhile mass in GR does not have a single definition that works perfectly in all cases. So I guess if we're talking about mass and assuming that GR is the best model of spacetime, we're necessarily mixing concepts that are not purely GR. But, we still use mass and can make sense of statements about it, I guess because there is no alternative that can better convey the same information as concisely and meaningfully.
Well, this is all rather pointless, except that I think the concept that energy in the COM frame is equivalent to mass is generally useful even if it can be problematic.
On 3/16/2023 at 5:42 AM, Markus Hanke said:I’m not so sure about this - the energies and momenta are certainly frame-dependent, but their sums should never cancel. Since E=hf, there is no physically realisable frame in which either beam is seen to have f=0 by the other beam, so I think you will always get a non-vanishing net energy of each beam with respect to the other. So I think in Newton, the beams should always attract according to an inverse-square law, irrespective of their relative direction of motion.
Sure, there's always energy in any given frame (even where it's too low to measure), but for beams with the same direction, none of it can be defined as rest mass, because there's no frame where the net momentum is zero.
However... if the beams aren't infinitely long, adding a single massive particle "to the system" then lets you define a rest mass that includes the beams. What would happen if you simply included a mass? My first intuition about it is certainly wrong.
With a mass, the COM frame doesn't depend on where the mass is. Would adding a small mass arbitrarily far away change the behavior of the beams? That certainly seems counter-intuitive. Or instead of a mass, you could just add another beam or pair of beams aimed in the opposite direction. If those were added very far away, would that make the 2 beams aimed in a common direction start attracting each other?
I think the answer is no, even though the system now has a rest mass. Indeed, if you have some beams close together in one direction, and some far away in another direction, that's approximately a description of the original experiment, except with a change of scale. The 2 beams aimed in one direction should bend toward the beams moving in the opposite direction, not toward each other.
So either intuition has broken down here, or the intuitive explanation is that it must be that when separate bits of energy combine to make up a rest mass, the density distribution of that mass is not necessarily the same as the density distribution of the energy that makes up it. The mass is located in some way between the oppositely directed beams, not inside the beams themselves.
(By this point, I don't know if it's a better argument that intuition fails, or that intuition can be improved to make sense of GR, but the standard explanation involving how the pp-waves of the different beams superimpose differently, might be a more intuitive and more useful way to think about it than this, and might better suggest what kind of mass distribution should be expected.)
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On 3/13/2023 at 3:33 AM, Markus Hanke said:
Yes, the motion does of course “contribute” to the geometry of this spacetime, as I said in my post - but not really in an intuitive Newtonian way of making it “stronger”, which is an ill-defined concept in GR.
I think my main confusion was taking this to mean that adding "mass" energy generally doesn't correspond with any "stronger" Newtonian gravity.
I'm also not interested in Newtonian gravity except as an approximation of the predictions of GR. But as you pointed out, in my example the strong frame-dragging effect has no approximation in Newtonian gravity. It should be possible to contrive an example where the Newtonian forces are small, and the purely GR aspects dominate and give results that are the exact opposite of what Newtonian gravity predicts.
20 hours ago, Markus Hanke said:As a simply example of where the Newtonian intuition fails completely, consider a particular spacetime called the Bonner beam - it’s essentially two parallel, very long beams of light. One is free to give each of these beams arbitrarily much energy. Because they contain lots of energy, and energy is equivalent to mass, we should be able to ascribe some notion of mass to them - meaning these beam should gravitationally attract one another, because of curvature etc. Right?
What you will actually find is that, if you shoot these beams parallel in the same direction, they will not attract at all; but if you shoot them in opposite directions (all other things equal), they will attract, but not according to Newtonian inverse square laws, but something much more complicated.
So, mass alone won’t always work well in GR - it sometimes gives the right intuition, but in other circumstances it can fail really badly.
Maybe, but in this example mass alone and the notion of energy having mass in the COM frame seems to work fine and fits with intuition. With 2 beams in the same direction, there's no COM frame. The energy of the beams is frame-dependent, and as total momentum approaches zero, so does the energy. Intuitively there is no energy that could make up "rest mass." With beams in opposite directions there is energy in the COM frame.
True, that's not enough to guess how much they'd attract. I wouldn't even be certain the beams would attract (because of the uncommon geometry of the "mass"). As you've said such a notion of mass would be "problematic" in GR. But I think at least it's intuitive that the energy that doesn't vanish with a choice of coordinates, curves spacetime around the beams, similar to how any other equivalent mass would (and I don't know how to say that without using the word 'mass' even if it's just in a vague way).
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I re-read the thread looking for why this is not at all making sense to me. One thing I see is I was only talking about curvature far away from the masses, while OP mentions only the curvature at the point they meet. Other replies talk about both, and it's only the stuff about curvature far away that's not making sense to me. Another thing is I might not understand what is meant by a "source of gravity".
I'll try again to rephrase OP's question to see if I can figure out where I'm going wrong.
Say that you have two identical massive objects far apart from each other, and a test particle at rest at the midpoint between them. Then say you add a relatively massive amount of energy to one of the objects by spinning it. A very basic understanding of relativity is that the test particle will fall toward the object with more mass (energy), and that it is due to curvature of spacetime that it falls.
Even though that additional energy is equivalent to mass, it is not what you're calling a "source of gravity"?
Does this experiment make sense with what's already been said in the thread? The only other difference I can think of is that I'm talking about a huge contribution of kinetic energy and maybe other replies are talking about insignificant amounts?
I'm also not sure if we're not in agreement about what would happen in an experiment like this, or if I'm just failing to understand the formal descriptions in GR.
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8 hours ago, Markus Hanke said:
In Newtonian gravity you can ascribe a gravitational force to each point of the surrounding space (iff you can define a potential field, which, btw, you can’t do here), which you can then compare - so if you analyse this in Newtonian terms, then the answer is probably yes. But in GR the question itself is essentially meaningless - all you have to compare are two metric tensors, and there is no meaningful way to say that one is greater/equal/less than the other. They are just different.
Then what is the correct way to describe their difference? Say you have 2 metrics, one for a system with little mass, and another for one with greater mass. The one with greater mass fits all intuitive notions of "stronger gravity", but how would that be expressed correctly? Greater mass corresponds with greater curvature, wouldn't the latter have greater curvature?
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Pirates
in Brain Teasers and Puzzles
Posted
My answer before isn't good enough.
Consider a worst-case scenario:
One pirate divides the loot, and the others evaluate. Say each of the other 4 think there is only one share that is less than equal, and that is the only share that they'd be satisfied with someone taking, but also each of the 4 think that a different share is the smallest. Then there is no share that can be kept by anyone, where they all agree on.
Therefore I think that no solution that involves the piles being split up, and then allowing one of them to be kept, can work. I think any solution that involves one person splitting up the loot, would require that the shares be adjusted before guaranteeing that they all agree.