What-If 0108: "Expensive Shoebox"

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Re: What-If 0108: "Expensive Shoebox"

Postby Quercus » Wed Mar 25, 2015 12:57 pm UTC

gmalivuk wrote:The rest of us are just trying to get an expensive shoebox.


An expensive shoebox, or an expensive shoebox sized volume? If you're firing rods of plutonium at stupid velocities into the thing in order to get them together before they go critical, I'm pretty sure by the time they're all in place you won't have anything which can reasonably be described as a shoebox anymore.

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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Wed Mar 25, 2015 2:03 pm UTC

rmsgrey wrote:
sevenperforce wrote:I don't think anyone is talking about doing a pure gun-type assembly. Because you can make the rods of plutonium you're firing toward the center of your "shoebox" arbitrarily many and arbitrarily thin, and give them arbitrarily high momentum...it should be doable.


If you're allowing arbitrarily high momentum, you could make a kinetic-energy bomb and not bother with the Plutonium... In fact you probably do better with something less dense since it'll have more kinetic energy per unit momentum.

I'm not interested in making a bomb; I just want to get all the plutonium into a shoebox at once, even if it's only for a split second.

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Wed Mar 25, 2015 2:34 pm UTC

sevenperforce wrote: Because you can make the rods of plutonium you're firing toward the center of your "shoebox" arbitrarily many and arbitrarily thin, and give them arbitrarily high momentum...it should be doable.
Nope. Not physically possible.

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Re: What-If 0108: "Expensive Shoebox"

Postby speising » Wed Mar 25, 2015 4:21 pm UTC

SFX wrote:
sevenperforce wrote: Because you can make the rods of plutonium you're firing toward the center of your "shoebox" arbitrarily many and arbitrarily thin, and give them arbitrarily high momentum...it should be doable.
Nope. Not physically possible.

[citation or at least explanation required]

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Re: What-If 0108: "Expensive Shoebox"

Postby gmalivuk » Wed Mar 25, 2015 5:02 pm UTC

There isn't one, because SFX is simply wrong.

The stored plutonium in the world is currently a safe distance apart, because it's not currently going critical.

Reactions, even nuclear reactions, take some nonzero time to happen, and energy takes a nonzero time to move from one place to another.

So if 300kg of the aforementioned plutonium were properly divided up and formed into the proper shapes, and was theoretically all moving toward the same point in spacetime at sufficiently high velocity, it would get to that point before any fission reactions had time to vaporize and disperse the stuff.

Sure, that's not a very *practial* thing to arrange, but it is absolutely physically possible.
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Wed Mar 25, 2015 6:19 pm UTC

speising wrote:
SFX wrote:
sevenperforce wrote: Because you can make the rods of plutonium you're firing toward the center of your "shoebox" arbitrarily many and arbitrarily thin, and give them arbitrarily high momentum...it should be doable.
Nope. Not physically possible.

[citation or at least explanation required]

Yeah -- for example, SFX could point to some equation that shows how increasing the incoming momentum causes a proportionally greater force once preignition starts, such that increasing velocity will never be able to overcome the forces during preignition.

For that matter, I'm not entirely certain it's been proven that a gun-type assembly plutonium nuke is theoretically impossible, as opposed to just prohibitively difficult to achieve (due to the necessary acceleration and so forth). If SFX is so certain, it should be easy enough to prove, right?

Consider the simplest possible case: two half-spheres of Plutonium-239, each massing 10 kg. The surface area of a 10 kg half-sphere is 362 cm2, 18% greater than the surface area of a critical-mass 10-kg sphere, so each of the half-spheres will be safely subcritical.

Place the two half-spheres at opposite ends of a nice long tube -- let's say 400 meters. The tube will be neutron-absorbent and quite strong, but it will have a series of openings at the target end to provide overpressure relief:

plut gntype.png
plut gntype.png (3.15 KiB) Viewed 9393 times


Pump the tube full of xenon-135 gas.

The half-sphere on the right will be accelerated leftward (using compressed gas or whatever) at some stupidly high speed. The xenon-135 filling the chamber will be expelled through the holes just in front of the target half-sphere, but will very effectively poison the criticality of the system right up until the two half-spheres are just inches apart. Should any molten or gaseous plutonium result at this point from preignition, they should be expelled out the holes as well due to the stupidly high momentum of the right-hand half-sphere.

With sufficient speed, the two half-spheres will be brought together with more than enough momentum to produce a prompt-critical configuration and undergo sufficient fission to surpass fizzle status. This, after all, was the basic design of the Thin Man plutonium bomb, though they realized that unexpectedly high plutonium-240 contamination made the design too unreliable. Perhaps if they had thought of pumping the damn thing full of xenon-135....

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Wed Mar 25, 2015 7:04 pm UTC

speising wrote: [citation or at least explanation required]
The resulting "weapons-grade" plutonium is typically about 93 percent plutonium-239 Pu-239 is the only Pu you can buy in quantity, theoretically. It can never be assembled into a large mass, like the shoebox, no matter how long a cannon you use, it makes a fizzle bomb before it reaches one place. This is for a very small amount. It is because of the Pu-240 that it is so impossible to get 300 kilos together like a shoebox size, in any way.

Side note, even 3 kg of Pu-239 will go critical in water, even with out a spherical shape.

Bottom line,
The plutonium-240 content even in weapons-grade plutonium is sufficiently large that very rapid assembly is necessary to prevent pre-initiation. Hence the simplest type of nuclear explosive, a "gun type," in which the optimum critical configuration is assembled more slowly than in an "implosion type" device, cannot be made with plutonium but only with highly enriched uranium, in which spontaneous fission is rare
. http://www.ccnr.org/plute.html

Now if you could get 300 kilos of pure Pu-239 somehow, (making it with a cyclotron, bombarding Uranium-238 with neutrons to produce uranium-239, which beta decays to neptunium-239, which then decays to plutonium-239, is theoretically possible, might take maybe a century, unless you built a huge amount of new cyclotrons), and you used really long guns, and very powerful explosives, it is theoretically possible to try shoot it all into one place, but it won't be a shoebox shape, and it will still go boom long before it all reaches the same place. But now we are closer to the theoretical jamming it all into a shoebox.

So theoretically you could almost fill a sort of shoebox area with pure Pu-239 for almost a microsecond, in which case you have to change the number for the money.

Because pure Pu-239 is really really really expensive. It takes weeks to make a microgram of it.

While you can never fill a shoebox with the 5000 dollars a gram weapons grade Pu-239, and certainly not reactor grade Pu-239, you could get close using pure Pu-239, but since you can't buy 300 kilos, that would make it priceless. You can't buy even a kilo of pure Pu-239, for obvious reasons. Anybody who could get their hands on one kilo of pure Pu-239 could easily make a gun type nuclear bomb. There are other serious problems however with using pure Pu-239 however. Getting 300 kilos together, the engineering for the huge gun set up would require a thousand cannons, at least, with all kinds of problems involved. But we are talking theoretical here.

Much of this is in the history of the nuclear bomb making projects.

The sources I used should answer further questions. (that was joke, of course they won't)

(EDIT) I see a post slipped in. Thin man link

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Wed Mar 25, 2015 7:11 pm UTC

sevenperforce wrote:Place the two half-spheres at opposite ends of a nice long tube -- .
If what you just posted is correct, you just told everyone on the planet how to cheaply build a plutonium bomb, with existing technology, so now anyone with 10 kilos of Pu can make a bomb.

I don't think that's a wise idea.

Since even 1 kilo of Pu-239 can make a very big explosion, and the gun tube would be only a half inch wide, your idea is sort of dangerous. Word to the wise should be sufficient.

See this link

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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Wed Mar 25, 2015 8:07 pm UTC

SFX wrote:
sevenperforce wrote:Place the two half-spheres at opposite ends of a nice long tube -- .
If what you just posted is correct, you just told everyone on the planet how to cheaply build a plutonium bomb, with existing technology, so now anyone with 10 kilos of Pu can make a bomb.

I don't think that's a wise idea.

You seem to think nuclear weapon design is somehow extremely complicated. It's not. Sure, a highly-efficient, low-fallout, ultra-high-yield design is quite complicated, but simply putting together a basic plutonium nuke is pretty easy. A prolate-spheroid linear implosion design without any sort of tamper can be used to make low-yield plutonium fission nuke with as little as 13 kg of weapons-grade 239Pu.
Spoiler:
Construct a hollow metal cylinder about a foot in diameter and put a 13-kg plutonium-gallium alloy prolate ellipsoid inside (the ellipsoid will be easy enough to construct because the gallium contamination will reduce criticality):
lin_impl.png
lin_impl.png (1.8 KiB) Viewed 9368 times

Stuff the hollow spaces with a neutron-absorbing high explosive and slap a detonator on each end. There you go. With some engineering tricks you can get the size down to under 6", as in the W48 nuclear artillery shell, but at a significantly higher materials cost. Of course no one is going to be able to do this without access to weapons-grade plutonium, which isn't particularly easy to come by.
Since even 1 kilo of Pu-239 can make a very big explosion, and the gun tube would be only a half inch wide, your idea is sort of dangerous. Word to the wise should be sufficient.

The gun-tube arrangement is not going to work at all with just 1 kg of 239Pu. Granted, you can make a tampered, neutron-reflected spherical implosion hollow-pit plutonium nuke with just 1 kg of the stuff, but that takes a LOT of engineering and physics know-how.

For a gun-tube design, you need 10 kg at theoretical minimum, but probably 15-20 kg. And you need weapons-grade stuff, not the reactor-grade stuff Putin gave Iran. And, for my design, you also need a bunch of xenon-135 gas. 135Xe has a half-life of only 9.2 hours so this is going to be completely pointless for any weaponized design; it would only be useful in the proof-of-concept I outlined. It's also (to my knowledge) the only gaseous neutron poison.

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Re: What-If 0108: "Expensive Shoebox"

Postby gmalivuk » Wed Mar 25, 2015 8:13 pm UTC

SFX wrote:Getting 300 kilos together, the engineering for the huge gun set up would require a thousand cannons, at least, with all kinds of problems involved. But we are talking theoretical here.
Yes, we are talking theoretical. You keep acting as if the rest of us expect this to be a practical thing to accomplish. Where did you ever get that idea?

SFX wrote:
sevenperforce wrote:Place the two half-spheres at opposite ends of a nice long tube -- .
If what you just posted is correct, you just told everyone on the planet how to cheaply build a plutonium bomb, with existing technology, so now anyone with 10 kilos of Pu can make a bomb.
Anyone with 10kg of Pu-239 could already make a bomb, with information that's already been publicly available for years, and they could make it in a much smaller space than 400 meters.
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Wed Mar 25, 2015 9:05 pm UTC

gmalivuk wrote:
SFX wrote:
sevenperforce wrote:Place the two half-spheres at opposite ends of a nice long tube -- .
If what you just posted is correct, you just told everyone on the planet how to cheaply build a plutonium bomb, with existing technology, so now anyone with 10 kilos of Pu can make a bomb.
Anyone with 10kg of Pu-239 could already make a bomb, with information that's already been publicly available for years, and they could make it in a much smaller space than 400 meters.

Given that the original question was "most expensive way to fill a size 11 shoebox" and he was considering various types of metal, I think you could manage to do it with slightly-contaminated plutonium if you chose the right contaminant.

A shoebox is going to be roughly 30cm x 18cm x 10cm. Make ten 239Pu plates that are each 30cm x 18cm x 1cm (each 30 kg) and coat them on top and bottom with samarium-139. Stack them in the shoebox. Walk away.

(Note: depending on the samarium-layer thickness needed to absorb enough neutrons, you may only be able to fit 7 or 8 of the plates into the shoebox.)

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Wed Mar 25, 2015 9:29 pm UTC

Yeah but using pure Pu-239 would be way more expensive. Like, I don't even know how to calculate the money value.

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Re: What-If 0108: "Expensive Shoebox"

Postby Neil_Boekend » Thu Mar 26, 2015 7:07 am UTC

sevenperforce wrote:Make ten 239Pu plates that are each 30cm x 18cm x 1cm (each 30 kg) and coat them on top and bottom with samarium-139. Stack them in the shoebox. Walk away.

Forgive me my ignorence, but what does Samarium-139 do? It doesn't seem like a good neutron absorber (like beryllium-9) because it just beta decays.
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Thu Mar 26, 2015 3:33 pm UTC

SFX wrote:Yeah but using pure Pu-239 would be way more expensive. Like, I don't even know how to calculate the money value.

Priceless, essentially, meaning it has no fair market value, meaning it can't be used to fill the shoebox in a meaningful way.

Neil_Boekend wrote:
sevenperforce wrote:Make ten 239Pu plates that are each 30cm x 18cm x 1cm (each 30 kg) and coat them on top and bottom with samarium-139. Stack them in the shoebox. Walk away.

Forgive me my ignorence, but what does Samarium-139 do? It doesn't seem like a good neutron absorber (like beryllium-9) because it just beta decays.

Oops -- I was reading the source wrong and assumed 139Sm was the second-best neutron poison after 135Xe, when in fact it is merely the second-best neutron poison which is produced by fissile reactions. Basic boron would almost certainly be the best neutron absorber. In fact, you don't even need to coat the 239Pu plates; you just need to stack boron plates between them.

But are you sure beryllium-9 is a good neutron absorber? All the sources I've read suggest that 9Be is a neutron reflector, not a neutron absorber. After all, beryllium spheres were used to for tickling the dragon's tail with the demon core.

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Re: What-If 0108: "Expensive Shoebox"

Postby Quercus » Thu Mar 26, 2015 3:41 pm UTC

Okay, so if the boron poisoning would work, the relevant question then becomes what is the optimal geometry to maximise plutonium to boron ratio and therefore expense?

And for extra credit, precisely how many government watch-lists will the googling to answer that question land you on?

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Re: What-If 0108: "Expensive Shoebox"

Postby Neil_Boekend » Thu Mar 26, 2015 4:30 pm UTC

No sevenperforce, I am not sure beryllium-9 would be a good neutron absorber.
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Thu Mar 26, 2015 4:46 pm UTC

Quercus wrote:Okay, so if the boron poisoning would work, the relevant question then becomes what is the optimal geometry to maximise plutonium to boron ratio and therefore expense?

And for extra credit, precisely how many government watch-lists will the googling to answer that question land you on?

Hah!

Probably none, because you won't find anything. This is the part of nuclear weapon design that's still classified, and will probably always be classified. Figuring out the precise geometry necessary for optimal nuclear weapon design requires you to combine extremely advanced theoretical physics with direct access to numerous test results from Los Alamos or similar labs, using mathematical abilities of the highest order.

If a rogue government or group managed to get its hands on some weapons-grade fissile material, it wouldn't be terribly hard to use it to cause damage. At the very least, such a group could grind the fissile material into shrapnel and pack it around a few tonnes of conventional explosive to make a radiological dispersal device, to devastating effect. Or, as SFX linked, 15-20 kg of weapons-grade plutonium could be placed in a Thin Man gun-type device with the full knowledge that it would fizzle (in comparison to true nuclear weapons) but still exceed its own weight in TNT-equivalent yield while maximizing fallout and radiological dispersal.

The only reasonable danger of rogue elements ever being able to produce a true nuclear bomb would be if they somehow got their hands on a couple hundred pounds of highly-enriched uranium (the type of uranium Iran has been trying to make for years). If that happened, they could construct a Little Boy assembly in any machine shop. In theory, it would be possible to build a true nuke out of 8-10 kg of weapons-grade plutonium by simply replicating the now-unclassified component design of the Fat Man or Trinity, but doing so would require access to a great deal of design and engineering and manufacturing expertise impossible without large-scale government funding. The North Korean government, which has access to ample weapons-grade plutonium and highly-enriched uranium from its dedicated reprocessing plants and centrifuge labs, was unable to manage more than a fizzle in its first nuclear test and only recently pulled off a 1-4 kt test.

Of course, in any of the above cases, it's delivery rather than design that poses the challenge. Crude nuclear weapons (or radiological dispersal weapons, either of the conventional-explosive or the nuclear-fizzle type) are huge, ungainly, and pretty much impossible for a rogue group to deliver to a target.

So yeah, none of that stuff is classified.

What is classified is the kind of stuff we're asking about in this thread, like the specific geometry to ride the edge of criticality, because that's what's required to manufacture highly-yield-efficient or ultracompact nuclear weapons. Like the W48 nuclear artillery shell that's just 6.1 inches wide. Or the 105-mm nuclear artillery shell that physicist Ted Taylor claimed is possible. A 105-mm nuclear device is just 4.1 inches in diameter...that's incredibly small, roughly the same diameter as a roll of duct tape.

That's why all those tests at Los Alamos and elsewhere were necessary (including tickling the dragon's tail). The physics of chain reaction generations and critical-mass geometry is incredibly, fantastically complex and subject to many, many unknown factors. It took decades of testing and significant loss of life for enough experimental data to allow the world's most advanced nuclear physicists to precisely predict critical-mass geometry.

So we're never going to have an answer to "what is the highest possible plutonium to boron ratio in a size 11 shoebox" because there are only a handful of physicists in the world who would be able to answer that question, even if they had unfettered access to all the classified experimental data.

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Re: What-If 0108: "Expensive Shoebox"

Postby Quercus » Thu Mar 26, 2015 5:20 pm UTC

Ha, that's brilliant (the shoebox conclusion, not the nuclear proliferation stuff - that bit's considerably less than brilliant):

Q: "What's the most expensive way to fill a size 11 shoebox?"
A: "That's classified information"

It's "need to know" that nobody needs to know...

This is why I love you guys so much. There's not many places one could ask that question and legitimately, with detailed explanations, get to that answer.

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Thu Mar 26, 2015 6:50 pm UTC

sevenperforce wrote:What is classified is the kind of stuff we're asking about in this thread, like the specific geometry to ride the edge of criticality,
Yep


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Re: What-If 0108: "Expensive Shoebox"

Postby Pfhorrest » Tue Apr 07, 2015 5:23 pm UTC

If you independently rediscover information that is in a sense "publicly available" (it's facts about the universe that you can experiment to find out) but obscure and dangerous and thus classified once the government first discovered it, can you get in trouble with the government for that?

Like, if the very existence of nuclear fission were a classified secret, say, and some physicist independently came up with the theory that it was possible, would he get a visit from some TLA?
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Tue Apr 07, 2015 5:50 pm UTC

Pfhorrest wrote:If you independently rediscover information that is in a sense "publicly available" (it's facts about the universe that you can experiment to find out) but obscure and dangerous and thus classified once the government first discovered it, can you get in trouble with the government for that?

Like, if the very existence of nuclear fission were a classified secret, say, and some physicist independently came up with the theory that it was possible, would he get a visit from some TLA?

It's possible. Not so likely nowadays just because of how easily information spreads.

If you use research of some kind to try and make public guesses about classified activities, the government is probably not going to comment or react one way or another lest they confirm your guess. Of course, if they think your guesses are too good and you must have some inside source of information, they'll definitely get a warrant and do a little snooping around.

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Tue Apr 07, 2015 6:27 pm UTC

Pfhorrest wrote:Like, if the very existence of nuclear fission were a classified secret, say, and some physicist independently came up with the theory that it was possible, would he get a visit from some TLA?
This almost exact event already occurred, during the worst possible time, WWII, while the Manhattan project was busy figuring out how to implode Pu-239 to make an explosion.

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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Tue Apr 07, 2015 7:07 pm UTC

SFX wrote:
Pfhorrest wrote:Like, if the very existence of nuclear fission were a classified secret, say, and some physicist independently came up with the theory that it was possible, would he get a visit from some TLA?
thith almost exact event already occurred, during the worst possible time, WWII, while the Manhattan project was busy figuring out how to implode Pu-239 to make an explosion.

With whom? Physicists had already come up with the idea of a chain reaction some time before the Manhattan Project started.

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Tue Apr 07, 2015 7:52 pm UTC

sevenperforce wrote:With whom? Physicists had already come up with the idea of a chain reaction some time before the Manhattan Project started.
It's a really cool story, and it might not even be online. I have to check, because I always have sources for posting.

Here's the really cool part of it. He went on to become famous, in large part from his figuring out what they were doing with Pu-239.

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Re: What-If 0108: "Expensive Shoebox"

Postby rmsgrey » Thu Apr 16, 2015 6:26 pm UTC

During the Manhattan Project, any research into atomic weaponry was kept highly classified, and the US government considered banning the publication of SF that included atomic bombs until someone pointed out that the sudden cessation of atom bomb stories would tip off a million or so SF fans, so the atom bomb remained the exclusive preserve of SF writers rather than serious science publications for some years...

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Fri Apr 17, 2015 3:11 pm UTC

It was Fred Hoyle. This source mentions it, but it's not the entire story.

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Re: What-If 0108: "Expensive Shoebox"

Postby SFX » Mon May 04, 2015 5:29 pm UTC

Pfhorrest wrote:If you independently rediscover information that is in a sense "publicly available" (it's facts about the universe that you can experiment to find out) but obscure and dangerous and thus classified once the government first discovered it, can you get in trouble with the government for that?
That is a question that can't be answered. But if you think about all goverbnments, not just the "good ones", of course you can get in trouble.

Think North Korea. Or Iran.

You could get in serious trouble for just discussing some things that are well known.

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Re: What-If 0108: "Expensive Shoebox"

Postby Pfhorrest » Mon May 04, 2015 11:49 pm UTC

I meant the US government implicitly since that seems to be the context. And I don't see why it can't be answered. I asked in in the context of "the specific geometry to ride the edge of criticality" being a classified secret, when I assume that that is something that could in principle be independently rediscovered, and since some people here seem to have firsthand experience dealing with classified things like that I wondered if they might know how the agencies responsible for keeping such secrets might react to someone independently rediscovering one.

It's not like the secret has to be revealed to know how the secret-holder would react to someone else stumbling across it.
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Re: What-If 0108: "Expensive Shoebox"

Postby sevenperforce » Wed May 06, 2015 6:19 pm UTC

The sort of people who would be capable of rediscovering those sorts of secrets would almost certainly have the knowledge and wherewithal to sell their secrets to the highest bidder if they so desired. If they weren't interested in selling their secrets, then they'd probably be interested in publishing. I assume that the government exercises some minute degree of censorial control over peer review so that certain classified secrets don't get published.

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Re: What-If 0108: "Expensive Shoebox"

Postby Keybounce » Sun Jul 16, 2017 8:54 pm UTC

Quercus wrote:Okay, so if the boron poisoning would work, the relevant question then becomes what is the optimal geometry to maximise plutonium to boron ratio and therefore expense?

And for extra credit, precisely how many government watch-lists will the googling to answer that question land you on?


I don't know how many watch lists I landed on, but the first result was this thread.

===

Reading this comic (sorry for being 3 years late), my first reaction to the idea of putting memory cards into a shoebox and filling them with songs is: You didn't account for the cost of the memory cards!

What would a shoebox full of memory cards cost? Heck, what would a shoebox full of M:tG power nine cards cost? (For that matter, what's the current set of cards banned from normal play for being too powerful, and what would that shoebox cost?)

===

Big numbers (was discussed in this thread, so ..): While 20^^^^20 may be big (quick question: Is there some summary notation, like 20^(4)20, to avoid having lots of arrows?), I understand that there are bigger things, and Ackerman is kinda tame. (If I understood it correctly, 1 = multiplication, 2 = power, 3 = height 2 power, 4 = height 3 power, etc -- and it gets there by adding "1" repeatedly, the successor function on overkill).

I used to think that 3^^^^3 arrows of uparrows was a lot, especially if you repeat that 64 times. But apparently a tiny forest is even bigger.

But value of that check? You could never cash it, so it has no value.
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gmalivuk
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Re: What-If 0108: "Expensive Shoebox"

Postby gmalivuk » Tue Jul 18, 2017 4:52 pm UTC

Keybounce wrote:(quick question: Is there some summary notation, like 20^(4)20, to avoid having lots of arrows?)
20^^^^20 = 20→20→4, using Conway chained arrows
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