Sometimes the lemonade you buy at the store is pink instead of yellow, because it has a different color of dye in it.
![PinkLemonade_2LBottle_Hero[1]](https://sfoil.files.wordpress.com/2018/06/pinklemonade_2lbottle_hero1.png?resize=270%2C480)
But what if you want to drink patrician lemonade instead of mass-market swill? Red food coloring is not the answer you’re looking for.
In the last post, I looked at what sort of damage North Korea can do with its Long Range Artillery, including both conventional and chemical payloads. Now we can try to estimate what the North can accomplish with nuclear weapons. The number and type of nuclear warheads and delivery systems the North Koreans possess isn’t knowable, but using some publicly available information and making up plausible-sounding numbers can give us an idea of what they can accomplish.
Effects
Presumably the North Koreans are confident in their 15kt bomb design, which they claim will (unlike the bombs detonated in 1945) fit onto a missile. There’s no reason not to believe them; if it doesn’t, they’ll just keep conducting tests until they have something that works and is small enough. If they don’t set off another 100kt+ detonation, it’s an indicator that the 2017 test was of a missile-suitable warhead, which it probably was – -they don’t have much use for something that can’t be put on a missile.
I’m going to use Alex Wellerstein’s NUKEMAP app to estimate the effects of a nuclear explosion, and as a comparison to the uniform population density method from the last post. He also has a “MISSILEMAP” app, but we’re not concerned with the ability to destroy a specific target; this is a countervalue attack against a densely populated area.
Playing around with the 15kt explosion on NUKEMAP, I get between 60-200k killed and 250-500k wounded depending on the nature of the district in which the bomb hits using Mr. Wellerstein’s model. In the .017人/m^2 uniform density model from the last post, and the 1200m 500 rem dose radius as a kill radius (this area also includes a shockwave >5psi), we get 76,867 fatalities. Using the 100% third-degree burns radius as the wound radius (1.9km) gives us 201,000 wounded. So, this is in the ballpark of Wellerstein’s more detailed model, on the low end. His model accounts for the possibility of casualties outside the radii noted above. Mine is much more conservative. I don’t put too much trust in the resolution of his population-density data, since the population shifts around the city substantially during the day and week. Still, we’re generally in agreement.
Commercial/Government vs Residential
When discussing the damage done by conventional munitions, I decided to reduce the “damage radius” in order to account for the protection provided by the broken, built-up urban environment. I don’t believe this to be the case with a nuclear detonation, because the large shockwave produced will tend to cause the total collapse of buildings, by applying the shockwave overpressure against a much larger area, compared to smaller conventional explosives. This isn’t an inherent characteristic of nuclear weapons, and emulating this effect with non-nuclear explosives is the point of weapons like the GBU-43 or Timothy McVeigh’s hobby project. However, none of the North Korean artillery warheads are anywhere near large enough to cause wholesale collapse of structures as in an earthquake. It is possible that the construction of larger buildings in Seoul would enable them to avoid collapse collapse by one or several nuclear blasts. If so, these are probably overestimates, but a lot of people are still going to die.
Aleppo, Syria in 2014. Despite heavy fighting, most of the buildings are basically intact.
Unfortunately, the paltry 15kt bomb is a firecracker compared to North Korea’s state of the art, for which I’ll use a conservative 100kt estimate. Depending, again, on the district hit, NUKEMAP estimates 245,000-480,000 dead and 800,000-1,200,000 wounded. Using our own model above — and 1600m/4500m kill/wound from the same source — gives us 140,000 dead and 1.1 million wounded. This is almost certainly lowballing the number of people killed, since many outside the radius would be killed by building collapse, burns, radiation, etc.
RIP Noise Basement
So, it looks like it would take 2-6x 100kt warheads or 5-15x 15kt warheads to inflict the same number of casualties as the conventional artillery barrage, while acknowledging that even one single warhead on target, especially the larger one, would do the job just fine.
Cost
We guessed in the last post that throwing 70 kilotons of high explosives at Seoul would cost about a billion dollars worth of munitions and require 50,000 people. The North Koreans’ job isn’t quite so simple as it appears above, because not every warhead can hit its target. If South Korean missile defenses can intercept 75% of incoming missiles, then the North need to launch 8 missiles to get a 90% chance of hitting. Even 50% interception rate means you need 3-4 missiles for a 90% probability of hit. Better, but do you feel lucky?
So, the North wants some combination of suppression and saturation of missile defenses in order to actually hit the target. If launching missiles in salvos reduces interception rates to 50%, and suppression of missile-defense sites (such as by maintaining a small fraction of the conventional artillery above, or by using small drones as improvised precision munitions) can get the interception rate to 33%, now we’re in business. Launch a five-missile salvo, 99% chance one goes off.
Nuclear disarmament advocate group Global Zero estimates that North Korea spends about $700 million per year on its nuclear program, compared to about $10 billion on its military overall. Estimating the marginal cost of a nuclear weapon is very difficult even in the best of circumstances. The 1998 book Atomic Audit estimates that China spent $28 billion in 1988 dollars ($58 billion in 2017) to build 450 nuclear warheads, about $120 million apiece. The comparison is apt given the similarities of a relatively poor state with modest aims. This article estimates a marginal cost of $50 million for an American ICBM. Given their possession of a workable warhead design and nuclear material, it is far from unreasonable to assume that North Korea can build and maintain 10x 100kt IRBMs and a flock of decoys for less than the billion dollar stock cost of its conventional deterrent munitions.
This also means that if nuclear forces allow the North Koreans to decrease spending on conventional forces by 10%, they’re coming out ahead. Spinning this shift as “conventional disarmament” even allows them to parlay the weapons program that in theory contributes to their pariah status into a diplomatic positive.
I read the first two books of Dan Simmons’ Hyperion Cantos (Hyperion and Fall of Hyperion). I have been roundly advised not to continue with series, and I am not, especially after reading the end of Fall. Overall these two books deserve their relatively high reputation.
Hyperion was the stronger book, I suspect because of its episodic nature. Both are around 500 pages long, but the density of the narrative in Fall is much higher since most of the space in Hyperion is taken up by the pilgrims’ loosely connected stories. Fall was good on its own (though you’re about to hear me complain about it some more), but just didn’t maintain the same high standard of quality. Any reader who likes Hyperion will find it very difficult not to pick up Fall to resolve the cliffhanger, anyway.
Recommended before reading this: If It’s Worth Doing, It’s Worth Doing with Made Up Numbers
North Korea last week offered to remove long range artillery from the DMZ. But of course: they don’t need it anymore. They’ve got nuclear weapons. But what, exactly, is the logic behind replacing the conventional deterrent with a nuclear one? After all, North Korea’s current deterrent system has worked.
North Korea’s long-range artillery (LRA) has up until now served as both a deterrent and as a deep-strike capability. The North Koreans have no hope of conducting any meaningful air attacks against South Korean targets during a war, so they emplaced the LRA instead. The artillery fire would be unobserved, but on the other hand most of the likely targets are static.
LRA represents a considerable investment in manpower and maintenance for the North Koreans. Their main delivery systems, in order of increasing range and power, are the 170mm “Koksan” gun, the 240mm multiple rocket launcher (MLRS), and the 300mm MLRS. The 170mm gun can reach the northern parts of Seoul, the 240mm rockets can reach most of Seoul, and the 300mm MLRS can reach down to Daejeon, threatening major installations south of Seoul.
I want to emphasize that this is a hypothetical to create an estimate for both the countervalue damage caused by this force, and give a rough estimate of North Korean investment. Accurate numbers for the number of these systems in use are not publicly available, if at all. Estimates are generally around “several hundred” 170mm guns, around two hundred 240mm MLRS, and fewer than a hundred 300mm MLRS. We’ll assume that all artillery is organized into battalions of 12 pieces each. We’ll go with 5x 300mm battalions (60 pieces), 15x 240mm battalions (180 pieces), and 40x 170mm battalions (480 pieces).
The Power of Long Range Artillery
Now we try to figure out how much damage these pieces can do. The Koksan is similar to the American M107 175mm howitzer, which was used in Vietnam. The M107 fires a ~150lb shell, at a rate of 2/minute, for 120/hr. The shell is about 50% larger than a 100lb 155/152mm shell, so instead of a 50m casualty radius we will go with a 60m radius (inverse square law in effect). To further specify, anyone within a 20m radius would be killed, anyone from 20-60m is wounded. This is for an airburst against a person standing in the open however, not in a heavily built-up urban environment. Relying mostly on intuition, I will quarter the casualty radius to make up for the natural cover provided by the surroundings and the tendency to seek immediate cover.
Assume the 240mm rocket is similar to the Russian 220mm Uragan, with a 220-lb warhead. We’ll go with a 30m kill radius and an 70m wound radius in the open, down to 7.5/17.5 with the terrain adjustment. Each launcher carries 22 rockets, which it can fire quickly, although it then needs to reload, reposition, and retarget. Perhaps one salvo every 20 minutes, or 22×3 = 66 rounds per hour.
The 300mm rocket is probably similar to the BM-30 Smerch, with a ~500lb warhead. 40m kill radius, 80m wound radius (adjusted to 10/20m). Each launcher has eight tubes, which it can fire in about a minute, although it takes a while to reload and reposition. Again, one salvo every 20 minutes, or 8×3 = 24 rounds per hour.
This gives us 57,600 170mm rounds, 11,880 240mm rounds, and 1,440 300mm rounds per hour. However, that’s an unrealistically high estimate. Not all of the launchers work, not all of the munitions will explode…and not all of the warheads will contain high explosive.
The museum-piece Koksans might have a 50% readiness rate and a 20% dud rate. They are now firing 28,800 170mm rounds, of which 23,000 will actually explode. The newer, better-maintained MLRS might have a 70% readiness rate and a 10% dud rate, giving us about 7,500x 240mm and 900x 300mm effective rockets per hour.
In an all-out scenario, some of those munitions will be fitted with chemical warheads. I’ll look at the effects later, but if I had to guess then maybe 10-20% of warheads would be chemical. Split the difference, call it 15%, and now we have 19,550x shells, 6,375x 220mm rockets, and 765x 300mm rockets exploding.
Since we’re considering this a countervalue strike, especially given the inaccuracy of these weapons, let’s look at Seoul. Seoul has a very high population density of 17,000 people per square kilometer, or .017 people per square meter. The 300mm rockets will probably be aimed past Seoul given their range, at relatively less dense targets like airfields and headquarters. I will use the town of Osan, just south of Seoul, as an exemplar; it has 200,000 people in about a 43 square kilometer area, giving us .0046 people per square meter.
In the first hour:
| Munition | Eff. Rnds | Kill Area (m2) | Wound Area (m2) | Pop/m2 | KILL | WOUND |
| 170mm | 19584 | 78.5 | 628 | 0.017 | 26,135 | 209,079 |
| 240mm | 6361 | 176.6 | 785 | 0.017 | 16,979 | 75,464 |
| 300mm | 771 | 314 | 942 | 0.0046 | 990 | 2,970 |
Total: 44,104 killed, 287,513 wounded
The ratio between wounded and killed looks a bit high to me. Perhaps it includes relatively minor injuries that wouldn’t require immediate treatment. A major hidden assumption, also, is uniform distribution of incoming shells: no round lands in the same place, and there isn’t even any overlap between casualty radii. Also, the firing batteries never run out of ammunition.
Once the first barrage lands, anyone in the target area will take cover. I will model this as a further reduction in the casualty radii by 75%. Also, counterattacks will begin; I will assume that these will reduce the attacking LRA by 1% per hour. Summing the resulting series gets another 136,400 dead and 890,000 wounded before the LRA is silenced.
Assuming the 170mm shell is 15% high explosive filler (as is the usual 155mm shell), or 22.5lbs, the total amount of high explosive launched at Seoul is about 70 kilotons. Keep that in mind!
Total: about 1.3 million casualties (180,500 dead, 1.1 million wounded wounded)
That’s just from high explosives, though. What about the chemical weapons?
The largest chemical attack so far has been Halabja 1988 in Iraq. At the time, Halabja looks to have had 70,000 people. The attack, using a modern cocktail of lethal agents, targeted the city indiscriminately, killing around 3,500-5,000 and injuring about 7,000-10,000. Halabja had and has a lower population density than Seoul overall, although it’s difficult to determine how much because most figures for the town clearly include sparsely populated surrounding areas. Also, the high-rise residences common in Seoul would provide some protection against the heavier-than-air chemical agents.
Ignoring the confounding factors, a simple ratio means another 500k-700k dead and 1 million-1.4 million wounded. Note that this would provoke the use of nuclear weapons by the United States in retaliation. I am here using an assumption that the munitions discussed above produce a “blanket” that affects the target collectively, rather than trying to determine the effects of each given impact. I believe this to be the correct approach based on my understanding of chemical warfare. I’m not that confident in the reliability of this number, but I think it’s a good benchmark.
(It looks like the most lethal effect of the explosives might simply be to drive the population into shelters where they’re killed en masse by chemical agents sinking down. People might be safer in their high rises; some accounts from the First World War noticed a similar dilemma, with chemicals lingering in trenches.)
The above gives us a total of about a million dead and two or three million wounded, many of whom would die later given the inability to treat so many people. These three to four million casualties are probably an upper bound on what the North can achieve with its conventional deterrent. For instance, it assumes no overlap between chemical and explosive casualties, which is silly, in addition to the absurd assumption of uniform distribution of munitions and generally zero overkill. However, even this worst-case scenario wouldn’t actually annihilate Seoul, much less South Korea. 80% of the population in the target area would survive, and two-thirds would be relatively unscathed.
The Cost
We assumed above that the North Koreans maintains 60 battalions of artillery. The American units with which I am most familiar have about 500 soldiers in a howitzer battalion and 250 in a rocket battalion. However, these units are expected to operate with a great deal more independence and mobility than the North Korean units being discussed here. If there are 200 soldiers in a North Korean LRA battalion, this gives us 12,000 soldiers. These are firing units, however; the total force is probably substantially larger, maybe 50,000.
Then there are the numbers of munitions. The forces above fired over 1.4 million artillery shells, 420k 220mm rockets, and over 50k 300mm rockets. Not a single one of these was fired in support of tactical maneuver. Even assuming minimal costs of $300/shell, $1000/rocket, and $2000/rocket, this is around a billion dollars in munitions, against a GDP of under $30 billion, before even attempting to actually fight anybody. The point is, it ain’t cheap. This didn’t even attempt to include the expensive intermediate-range Scud-type missiles in the North, which are probably counterforce weapons, more or less.
Next, I’ll look at what why nuclear weapons might be an attractive replacement for all of this — which is what I originally intended to write about — in a follow-on post.
Edit: After I posted this, I found a much more detailed/competent/professional analysis of this problem by Mr. Roger Cavazos. I’m encouraged that my first-volley estimate was at least in the ballpark of his more informed method.
I finally got around to watching Andrei Tarkovsky’s 1972 Solaris. I enjoy “deliberate” movies — Barry Lyndon is one of my favorites — so I was not bothered by the slow pace, but most viewers will be. The only part of the movie I’d ever specifically heard of — a digressive scene of Tokyo traffic which length Tarkovsky may have intended solely to justify the Motherland’s expenses in sending his crew to Japan — was a little awkward but the rest of the movie was sublime. Lem’s complaints about the adaptation are unwarranted; the director simply had a different vision in a different medium. Imagine Clarke griping about Kubrick’s film (although Lem’s novel is superior to Clarke’s 2001).
On War, along with other memoirs and histories of its era, concerns itself in large part with fortresses. Large, partly self-sustaining fortified areas (food production was always impossible) continued to be important to military strategy even after cannon brought down the old castle walls.
Fortresses had several characteristics:
They controlled strategically important terrain, such as roads, mountain passes, and riverways. Typically, these features passed through the fortress.
They were large enough to house substantial bodies of troops, materiel, and supplies, well in excess of the permanent garrison.
They allowed armed forces inside to sally and return at will either en masse or in small groups.
They were sufficiently protected, both by fortification and armament, to require a concerted “main effort” to reduce. Successfully storming the fortress (in modern terms an “assault”, though “storm” is still used in German) required the most effort; an “investment” or siege to prevent sallies and eventually starve out the garrison a lesser effort. An “observation” or screen could be placed around a fortress with even less resources, although this would still allow entry and exit from the fortress more or less at will. In the 17th century, an entire field army might be required to reduce a fortress; as armies grew larger this became less the case, but fortresses remained an important feature while declining in importance compared to field fortifications through World War I.
What Happened?
There are two culprits for the decline of the fortress: mobility and firepower.
First, the increased mobility of both mechanized and air-dropped ground forces made it much more difficult for static fortifications to control terrain features. Motorized and aerial supply lines also make it much more difficult for a bypassed position to disrupt the communications of the bypassing unit.
However, the increased range, power, and precision of modern weapons clearly has the most to do with the decline of the fortress. Air-dropped munitions can easily destroy most fortifications, and modern munitions will hit whatever can be detected from outside the range of defensive armaments. The last identifiable fortresses — such as Malta — relied in the final assessment on the inaccuracy of aerial bombs to survive.
What would a Modern Fortress look like?
A modern fortress would have the characteristics described above, but I will add an important qualifier:
It should cost more to destroy than to build.
I won’t get too wrapped up in this, but: if our fortress takes a great deal of effort to create, but can easily be eliminated by a readily-available, inexpensive weapon or attack, it’s not what we’re looking for.
I’d also like to add something else in to make it more specific, based on the concept of “surfaces and gaps”. A surface is, basically, a strong point (not necessarily a physical location) and a gap is an exploitable weakness. A surface or gap does not need to always be such to everyone all the time. An air defense battery presents a “surface” to an aircraft (when it’s turned on!) but a gap to an infantry platoon.
A fortress should be “all surface” to an attacking force.
With this in mind, what are modern equivalents?
Airbases
Both the Donetsk War (Donetsk Airport) and the Syrian Civil War (Menagh, Al-Tabqa, Al-Duhur, Kuweires) have seen airports and air bases hold out against enemy attack long after surrounding territory. All of these sites had protective perimeter fortifications, and held out far longer than the surrounding territory in the face of enemy attack (in the Syrian case, for several years). Donetsk did not last as long, but was also not as extensively hardened.
Menagh Air Base — The defending SAA occupied the airbase, while attacking rebels operated in the town to the northeast.
Donetsk Airport in early 2015
One thing that all of these had in common, though, are the lack of an opposing air-to-surface threat. While the enemy was reasonably well equipped in both cases — including artillery and armor — they had no ability to attack these installations from the air. In Syria, the government was able to keep the defending forces resupplied via helicopter for several years until defeat or, in the case of Kuweires, relief.
Forward Operating Bases
Expeditionary Forward Operating Bases (FOBs) have many similarities to the airbases above. However, they are purpose-built to actively support ground tactical operations outside their perimeter, as the examples above are not. They do include both perimeter fortifications and point-defense systems to destroy incoming standoff munitions. American “air defense” at bases has mostly been focused on point defense against crude artillery. Additionally, they may deployed integrated air defense systems (IADS).
Camp Bastion in Afghanistan
Typically an IADS is conceived as protecting an area, but this needn’t be the case. The Russian military base in Syria at Khmeimim is clearly the protected objective of such a network. Unlike the typical American FOBs, the Russians clearly consider a sophisticated aerial attack a real possibility. In accordance with the “all surface” condition above, the defense system shouldn’t require any component outside of the perimeter defense, although this doesn’t preclude the positioning of air defense sensors outside the perimeter, nor the interlinking of the base with a larger network (as indeed is implied by the very term IADS).
Point defenses, however advanced, could almost certainly not prevent a hit from an ICBM or other suborbital attack, but anything short of that remains an open question. Currently, these air defense systems appear key to the creation of fortress-type installations.
I think it worth noting that any FOB worth its salt has at least limited facilities for air transport, and larger ones inevitably have full-service airstrips.
Tunnel Complexes
On organization that didn’t have the benefit of air superiority was the Viet Cong. Their most extensive tunnel complexes, I believe, meet the definition of a fortress given above. The most famous example is at Cu Chi:
The tunnels were by all accounts nowhere near as cozy as this illustration implies
This differs from a simple bunker in enabling stockpiling of supplies, providing assembly areas, etc. Being underground provides both protection and concealment. The camouflage aspect is most interesting, I think: if every everything that can be seen can be hit, and anything hit is destroyed, then you must not be seen.
Many of the tunnels — including the Cu Chi headquarters — were eventually destroyed. However, this was not done without an intense, concerted effort, and the tunnels operated for years on end.
The South China Sea Islands
The Chinese installations in the South China Sea (such as the one below in the Spratly Islands) may be the closest modern equivalent to a fortress. Like the airbases and FOBs above, they have clear fields of fire in all directions (being islands); this is a common feature except for the tunnel complexes. They’re also large enough to allow critical defense systems some room to move around on the island; only the runway itself is completely fixed. This need for some local dispersion is also common to the airbases above, and to some degree is even present in the tunnel complexes, which could dig out new tunnels or shift into new areas as needed.
Chinese installation in the Spratly Islands (Fiery Cross Reef). The runway is about two miles long.
Cities
Important towns and cities are often located on strategically significant terrain, and may constitute such terrain themselves. As a result, fortified towns and cities are nothing new. Unlike the previous examples, seamless perimeter defense appear no longer feasible. Like tunnel complexes, their defense generally allows for some limited penetration by an attacking force.
Cities are now much larger, relative to armed forces of any size, than in the historical past. This makes the defense or investment of a city a much more fluid affair than in the past, where city walls and a complete countervallation were standard features of warfare.
21st-century cities are a challenging military problem, but I believe they are better conceived as a complex sort of terrain rather than as a “fortress”, which is fundamentally a “point” position. An air-defense network could easily cover a large city, but the city itself strikes me as too permeable to enemy attack to constitute a fortress. However, the terrain of a city could be conceived as allowing for a “fortress”-like strongpoints, providing camouflage. Such strongpoints are ubiquitous in urban warfare.
The thought of a modern city truly converted into a true fortress is intriguing. The defender would have to construct a perimeter defense, and the garrison would almost certainly require the mobilization and arming of the city’s inhabitants in an organized manner. The coordination of the defense would be an incredible undertaking that would probably build on existing law-enforcement and governance structure. The mass-mobilization aspect runs against current trends towards smaller, professional regular forces; the garrison would almost certainly be outside of the “regular” armed forces structure. Vague gestures in this direction are sometimes background in science fiction, but nothing like it exists in the real world. An interesting concept.
Conclusions
The ability both to defend against aerial attack, and to enable the defender to conduct at least limited air operations, looks like a key capability for a modern fortress. Air defenses have been improving — the ability to intercept both low-flying cruise missiles and high-flying aircraft. Point defense systems might actually encourage air defense components to be placed more closely together (by reducing the threat of HARM weapons and the like), encouraging perimeter defenses. A look at the images above suggests that some level of dispersion inside the “fortress” is another key defensive capability, giving “garrisoned” artillery and air defense systems a space to move around against counterfire. Defensive dispersion and space for operating aircraft appear to be synergistic: you need a square mile or few to conceal the locations or your important defense systems, so you might as well have a runway in there.
The necessity for perimeter barriers and clear fields of direct fire (the needed range extended by modern weapons) remains. The Chinese installations in the South China Sea clearly has these concepts in mind, as do Russian-operated bases in Syria.
Inspired by a trailer for a terrible-looking movie
Eev filed out of the dormitory and sat down to eat with the others, rubbing both eyes in an unthinking attempt to wake up just a little more as the noise of the eating hall hammered at her skull. The damp, nutty scent of her mycobroth did little to aid the process. Miky settled down next to her.
“Eat up, or you’re going to be hangry by lunch again,” Miky chided. Eev lolled her head to the right, swept a dark, corkscrew ringlet from her eyes and regarded her younger brother. A few years younger than her, but just as scrawny, and with the same smooth, nut-brown complexion and fine features even though he wasn’t, biologically, “related” to her – so far as she or anyone knew, at least. She grunted acknowledgement but Miky had already turned his face down, intent on slurping his broth before the end of the eating period.
Eev herself only spooned down half of the thick broth before the bell sounded. She stood up from the bench and followed her dorm neighbor, Alc, out of the concrete eating hall. She saw the overcast sky through the high windows but paid no more attention than usual to the banners hanging below the national raven insignia, proclaiming Village 13’s past accomplishments in contests of sporting and intellect – both areas in which Eev excelled, although she wasn’t old enough for individual recognition.
President Trump will meet with Kim Jong Un on Tuesday in Singapore. Nothing concrete will come out of this summit — the lack of South Korean representation guarantees that — but if things go well some sort of agreement is possible eventually. Still, both sides have things to offer economically, diplomatically, and militarily. What might that look like?
The alien ocean in Solaris remains beyond human understanding at the end of the novel. The ending of Neon Genesis Evangelion is also beyond understanding; this is not a coincidence.
In a previous post, I talked about what parts of Carl von Clausewitz’s On War I thought had little relevance today. Let’s look at what has endured. Once again, I’ll start with the most difficult concepts and save the more trivial or obvious observations until the end.
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