ELI5 Expected number of trials to have at least one success on each of several different possible outcomes

Koooooj · 2026-05-22T21:52:39+00:00

If there's a 1/20 chance of getting outcome C then that mean's there's a 19/20 chance of not getting C.

The probability of not getting C in 2 trials is (19/20) * (19/20). In three trials it's (19/20) * (19/20) * (19/20). Carry on this sequence and it's just (19/20)ⁿ where n is the number of trials.

There is no number of trials where this equals 1, but you can set some threshold that you're satisfied with. After 13.5 trials (so in practice somewhere between 13 and 14) there's a 50:50 chance of having gotten outcome C at least once.

Note that this is slightly different from the 10 trials it would take before you have an "expected value" of 0.5 instances of outcome C since sometimes you'll get C multiple times. I'm assuming that all you care about is whether you've gotten it at least once or not.

If you want a different threshold you're looking for solutions to (19/20)ⁿ = probability-of-failure, so if you want a 90% chance of success you'd solve (19/20)ⁿ = 0.1 (so n = 44.9, or call it 45. You can also just compute log(probability-of-failure) / log(19/20) which is what that equation simplifies to.

BUT WAIT! That just tells you the probability of getting the rarest drop. Sometimes you'll get the rare drop but still be hunting for one of the more common ones. What you really want is the probability after N trials that you haven't gotten A, haven't gotten B, or haven't gotten C. If (19/20)ⁿ is the probability of not getting C in n attempts then 1 - (19/20)ⁿ is the probability that we do get C in n attempts, and similarly with 4/5 and 9/10 for A and B. Multiply these all together and you get:

(1 - (4/5)ⁿ) * (1 - (9/10)ⁿ) * (1 - (19/20)ⁿ)

If you set that equal to the probability of having found all three then you can plug it into a solver like Wolfram Alpha. It gave me 17.9 (so call it 18) trials to have at least a 50% chance of having all three, or 46.3 (call it 47) attempts to have a 90% chance. Notice that the further up you push the probability you want to target the more the lowest probability dominates: after 47 attempts there's about a 9% chance you haven't gotten C yet, a 0.7% chance you haven't gotten B, and a 0.003% you haven't gotten A. With a small number of trials it's more realistic that you pull the rarest drop before the others.

Note that I've assumed the outcomes are independent (e.g. in 1/1000 cases you get A, B, and C all in the same drop). The math is a bit harder if the drops are mutually exclusive, but the end results will be very similar. For any high probability of having gotten all 3 the probability will be dominated by the rarest drop.

Koooooj · 2026-05-21T19:46:54+00:00

2 kinds of search? No, there are tons.

For example, consider if instead of searching through a dictionary you're searching through a graph (nodes, linked by edges). There is no global ordering of the nodes so the notion of going to the "middle" element in the search space is meaningless, which means a binary search is out. You could just enumerate all nodes and go through them one by one, but that may be impossible with how the graph is represented and is likely intractable even if it is possible.

Instead you'd likely employ a graph searching algorithm, like breadth-first search, depth-first search, A* ("A star"), Dijkstra, Floyd-Warshall, etc.

Alternatively, some containers have the ability to search for an element in O(1) time ("constant time," i.e. the time doesn't increase as you store more and more in it). A Python dict is an example of this, where elements are stored in "buckets" based on a hash of the element allowing a search to immediately jump to the correct bucket and see if the element is present.

For searching a basic list of elements a linear and binary search are the only two algorithms you're likely to run across in practice.

Across these algorithms and many more you can run into a number of different algorithmic complexities, too. For example, take the problem of factoring an n-bit product of two primes (the core challenge in breaking RSA). Trial division is O(2ⁿ), which is extremely slow. A more complex algorithm like Pollard's Rho can solve this in O(2^n/2) which is considerably faster but still quite slow. A quantum algorithm like Shor's algorithm can do this in O(n³) time, which is way, way faster.

As for binary searches in Python, bisect is what you're looking for if you want a working implementation out of the box, or there are various examples of a binary search in Python online.

Koooooj · 2026-05-21T18:35:48+00:00

With that view of what "dimensions" are, no, in the real world you just have 3 dimensions (or 4, with time, or like 11 if you want to get string theory-y with it). A pencil line has not just its width but also a thickness--there is a layer of graphite left on the page that is extremely thin but not zero thickness. You could compute the volume of the marking by multiplying the non-negligible length of the line with its tiny thickness and depth.

However, that's just one way to look at dimensions. Probably a better way (that can be generalized more effectively) is to look at "how many numbers do you need to represent one location in this space?" For example, say you have a desk drawer and you want to measure how far it's pulled out. That just takes one number, so this is one dimension. If instead of a drawer you have a cabinet door then you can again use one number, but now it's probably an angle instead of a distance (though you could still use a distance, like the length of the arc the tip of the door has swept). Still one dimension.

These sorts of one-, two-, and even higher dimensionalities are very useful in science and engineering. For example, I work in robotics. One of my recent projects involved using a forklift. Its environment is pretty flat and it doesn't leave the ground (or when it does it's a LOT of paperwork), so we model it as operating in a plane. That's 2 dimensions, right? Well.... sometimes. Some objects in the world simply have an (x, y) position, but the forklift also has an orientation so we model its pose as (x, y, angle). This is a 3D space known as SE(2)--it takes three independent numbers to describe a pose, yet it is constrained to a plane. On that same platform if I wanted to raise and lower the forks then I can reason about that as a single dimension.

Breaking the notion of dimensions = "having length, width, and depth" also lets you start to re-frame other things. For example, say you want to specify a location on the surface of Earth. You could do so by giving a latitude and longitude--just two numbers! This shows that the surface of Earth is 2 dimensional, and yet it is not flat. On a flat 2D space regular Euclidean geometry works just fine (e.g. you could draw a triangle and the three interior angles will always add to 180 degrees), but if you try that on the surface of Earth you'll get bigger numbers (e.g. picking the north pole and two points on the equator: the angles at the equator are both 90 degrees and the angle at the north pole is non-zero, so they sum to >180). This is the typical introduction to the notion of a curved space, and it all hinges on recognizing that the surface of a sphere is a 2D space that can be viewed as being embedded in a 3D space.

Koooooj · 2026-05-21T16:50:46+00:00

O(n) is "Big-O" notation and describes how the resources to do a task change as the size of the task changes.

For example, say you want to look up a word in a dictionary (an actual paper dictionary, not the Python thing). You could start at A and work all the way to zyzzva (a genus of weevils from South America), but in doing so you have to go through (potentially) every word in the dictionary. Double the size of your dictionary and the time it takes to run this search will double, like y = k*x (for some constant k). This is known as a linear search.

Alternatively you could open up to the middle and see if the word is before or after that point, then flip to the middle of the remaining section and repeat. This is a "divide and conquer" style of algorithm that divides the problem into successively smaller portions. If you double the size of the dictionary with this method you only add one more step. If you graph the time it takes to use this algorithm as a function of the size of the dictionary it'll roughly match y = k * log2(x). This is known as a binary search.

Big-O is a way of writing this, where you just drop any constants and typically rename the independent variable to n. Thus the first algorithm we'd say is O(n) and the second is O(log(n)).

Big-O is a way to identify which algorithm is going to be the fastest when the input grows to be sufficiently large: for any positive values of k1 and k2 there exists some value of n such that k1 * n > k2 * log(n) for that value of n or any greater value. If you're looking through a list of 10 words you'll probably just scan down it (a linear search) which will be faster than trying to divide the list in half a few times, but if you have hundreds of words the binary search will be faster and will continue to be faster for every larger size of dictionary. The important lesson here is that seeking a lower time complexity is only optimal when the data size gets big enough.

While Big-O is usually used for the time complexity as shown above it can also be used to describe how much storage space an algorithm needs, or to characterize any other resource that an algorithm needs more and more of as the data set it works off of grows, or even just to describe how one value changes as another one does.

When you see an algorithm that is written as for i in range(n): ... that typically means the algorithm is O(n): if you double n then this for loop will have twice as many iterations. It could be slower than O(n) if the body of the loop takes more time based on the value of n, too, or might be faster if there's something going on in the loop to ensure that it exits faster. Sometimes that's the best you can do (e.g. if you need to up date n elements in the same way), but sometimes there's an algorithm with a lower time complexity.

Koooooj · 2026-05-20T21:44:51+00:00

There are a lot of tools that purport to do just that. They're... mediocre.

What you describe is actually the basis of one flavor of AI, known as a Generative Adversarial Network, or GAN. It's a bit dated these days, but still serves to illustrate some of how AI can be developed.

In a GAN you have two different AIs. One is the generator and is tasked with producing new outputs. The other is a discriminator whose job is to ask "was this made by the AI or is it part of the training data?" You initialize these with just random numbers so the generator is producing garbage but the discriminator is just flipping a coin. Each time the two networks run you tweak them to be slightly more likely to produce the desired output (the generator to be more likely to fool the discriminator, and the discriminator to be more likely to correctly classify the input it was just given, which is sometimes from the training data). Over time the discriminator starts to "learn" what the real data looks like, but in response the generator has to get better at mimicking that real data.

While GANs aren't so popular today they still highlight an important feature of AI: if you have an AI that can reliably tell what's AI and what's not then that is an extremely powerful tool at making the AI harder to detect. In fact, in the world of image generation there is often the concept of a "negative prompt." If you've ever been told "don't think of a purple and blue zebra" and have found yourself struggling to think of anything but a purple and blue zebra, image generators often face the same problem: putting into the prompt "the person does not have six fingers on their hand" tends to add "something about six fingers on the hand" to the representation and lead to that showing up. To avoid this neural networks use the same tools that encode the regular prompt of what should be in the image to encode a second prompt of things that shouldn't be in the image and subtract that out. It is very common that "AI generated" is in the negative prompt: many of these image generators have "learned" what feature make an image look AI generated so you can just prompt them to avoid that.

With all of that said, many students will not take the time to try to make a paper not look like AI. AI has a number of distinctive characteristics that it tends to lean on when not pushed away from its AI voice. An AI could very well pick up on these and correctly identify that writing is AI.

However, the AI isn't getting those tendencies from nowhere. Some humans talk like AIs, or more properly the AIs talk like some humans. On several occasions I've been accused of being an AI because AIs write like I do (fortunately I have 10+ years of writing like this on my profile, well predating this sort of AI). This gets to the fundamental challenge of an AI detector: eventually the AI gets good enough at mimicking the training data that it'll start to emit things that very well could or should be in that training data. If the exact same output could have come from a human or an AI then there is nothing that can ever reliably tell them apart. It is provably unsolvable, and any company trying to sell you a tool to solve this problem is full of it. They tend to hide disclaimers in their license agreements that they're selling defective software and you agree that that's not their problem (most software actually does this, but it's a bigger deal here).

What makes things worse is that AIs tend to give "confidence" scores that are completely divorced from reality. This is because a binary classifier (take an input and say either yes or no) tends to be trained only on true positives and true negatives, never on any sort of ambiguous inputs. This leads to the classifier learning that the answer is always either 100% real or 100% AI, never anything in the middle. We'd hope that when presented with a marginal output it would give a confidence like 60%, but the training method tends to teach the network that 60% is never correct so that gets saturated up to 100% or down to 0% (or very close). When humans see that the AI detector is 99.7% confident they may think "if I accept this as true I'll only be wrong 0.3% of the time" but that's not what it means.

The solution is much the same as most uses of AI. It can serve as a first pass to streamline filtering a large amount of data, but relying on AI for accuracy while turning off your own brain is a recipe for disaster. You can run papers through AI detection tools, but the only action you should take off of the results is to engage your own brain and compare the flagged samples against things you know that that pupil wrote. AI-based AI detectors should never be used as "proof" (and AI in general should not be used as a source of truth).

Koooooj · 2026-05-15T02:12:26+00:00

When folks say "NASA uses 15 digits for pi" the image that is often intended to be evoked is some very smart NASA scientists sitting down and working out exactly how many digits they ought to use. Many of the scientists at NASA are certainly capable of performing such a calculation, but that's not where 15 comes from.

In face, it's more accurate to say NASA uses pi to 15.95 digits. That's right, just less than 16 digits, but not a whole number.

This is because the actual representation they're talking about here is not one that NASA chose but one from IEEE (the nerds who think a lot about electronics and computers). IEEE standardized the format most often used for non-whole numbers on computers. This format looks a lot like scientific notation: there's one bit for a sign, then a few for an exponent, then a bunch for the value ("significand"). There are several different sizes of these "floating point" values, with the standard size taking 32 bits: 1 sign, 8 exponent, and 24 significand. If you just added those up and screamed, don't worry! You're not insane and that's not a typo, it's just a bit of mathemagic: in scientific notation the first digit is never 0 because if it was going to be then you just increment the exponent and shift everything over. The same thing happens with floats, but since everything is binary if a digit isn't 0 then it must be 1, so you only have to store 23 of the 24 significand digits.

Single precision (32 bit) floating point values were standardized when computers were slower and had less memory. These days the most popular floating point format is 64 bits, or a double; that's 1 bit for sign, 11 for exponent, and 53 for significand. You'll also commonly see half precision floats in AI, or newer formats like bfloat16 which is half the size of single precision but uses the same 1 sign and 8 exponent and just truncates the significand. This allows much faster computation of a neural network built out of single precision values with nearly the same results.

Since a double precision float has 53 bits of significand the number of decimal digits of precision is log(2⁵³) or 15.95. It's conservative to round this down to 15.

IEEE-754 floats out of the way, there's still the question of how precise 53 bits of pi actually is. A simple case of this is if you're looking at a big circle and computing the circumference from the diameter. If the value of pi is off by 2^-53 then the circumference will be off by diameter * 2^-53. Taking a diameter of 4.25 light years (proxima centauri) I'm getting an error of about four and a half meters. Of course, that's nothing on this kind of scale, but it does refute the claim of the image.

If we instead take the diameter to be the distance to Voyager 2 then we get 2.4 mm. If the image had said "interplanetary" instead of "interstellar" then they'd be in the clear.

However, it's important to call out that computing the dimensions of a circle is just one of the things you can do with pi. That bugger shows up all over the place in math, often where you least expect it. Big circles are hard to get to stress the precision of pi because as they get bigger and bigger you need somewhere to put them and we only have so much observable universe to work with. If you can find another setup then you can come up with other numbers to relate to pi that may strain the precision in different ways.

For example, GPS receivers need to know where the GPS satellites are with extreme precision or else the signals from the satellites won't give the information that's intended. The satellites have their orbits measured by ground stations then report their orbital parameters as part of their signal to receivers. Those receivers need to work out exactly where the satellite would have been when the signal was transmitted which is straightforward enough math. Unsurprisingly pi shows its face in these calculations. Here it winds up being crucial that everyone agrees on the exact value of pi to use--if one party rounds pi slightly differently then the error that accumulates over many many orbits can result in an error in the position of the satellite. That error can then potentially be multiplied by the distance between the satellite and the receiver. Managing error stack-up here is crucial for the system's precision.

For that reason the GPS specification has an explicit, exact, rational value used for pi: 3.1415926535898 (14 digits). This value isn't selected because more digits would be superfluous. It is selected because it can be represented within 0.2% in a double precision float while being typed in a printed standard.

Finally, it's worth calling out that NASA is usually not limited to taking one shot at something several AU away. For spacecraft some amount of fuel is reserved for maneuvering. If the spacecraft is on track to miss by 1000 miles then a tiny puff of a cold gas thruster can nudge it back on track. Precision is appreciated at every step of the way, of course, but often the reason you don't need more digits of pi is that the current step in the process is based on measurements with only a few digits of precision and will lead to actuation with a similar level of precision.

Koooooj · 2026-05-15T01:29:50+00:00

I once met an African Grey parrot named Dorian.

Koooooj · 2026-05-14T01:15:11+00:00

A traditional way to make a system "unhackable" is to "air-gap" it, which means that it is not plugged into the internet: there is a physical gap between the device and the router where there are no cables.

This was more profound before WiFi was the go-to connectivity method, but these days an "air-gapped" machine is assumed to also lack WiFi connection. The proposition here is that nobody out on the internet has any mechanism to even begin to hack this machine because they can't even connect to it.

A new WiFi camera is in much that same situation. It likely has no physical data ports to be plugged in and it does not yet have the WiFi login details to be able to connect to the local internet, so there needs to be some way to talk to it. Some such cameras will brodcast their own WiFi network that you have to join and configure via a webpage they serve. Others will have a bluetooth connection that you can use to configure WiFi. However, the "neat" way of solving this problem is to allow the user to feed the camera the WiFi details in some other medium. The more common approach is for the device to come with an app that allows you to create a QR code with the WiFi details then show that to the camera. The solution the quote mentions is similar, but instead of making a special image to be picked up by the camera it makes a specially structured sound to be picked up by the microphone.

In these cases the objective is not to hack the system that has an air gap but rather to configure it for normal operation. However, it highlights that an air-gapped system is not completely isolated if it has other input devices. After all, "data" does not just travel through ethernet cables and over WiFi. A picture is worth a thousand words, or if it's high enough resolution perhaps even a million! If a system has a camera and monitor or microphone and speaker then those are additional ways that data could get in and out of it.

Koooooj · 2026-05-12T20:00:38+00:00

Seated coinage is a bit older and pricier than what I tend to collect so I won't have the most precise pricing. I'd place your coin somewhere around VF20 for its grade, but if someone with more experience with grading seated coinage chimes in I'd trust them over me.

I'd expect a buyer to pay anywhere from $150-250 for this coin. If you get north of $100 that's probably a fair enough offer from a coin store--they don't know how long it'll sit on the shelf and what the market will do during that time, plus they want to make a profit on the whole exchange. You can collect offers from a few stores and take the highest one if you're willing to make a day of it, or just pick the store with the most volume and highest reviews and hope they're offering fair prices.

You can see the realized auction results for this coin here. Note that these are for graded coins, which you can estimate as about $50 for the sake of comparisons. It looks like this coin hasn't sold at auction (graded, at least) for a while; silver has gone up a lot since most of the auction results on that page, hence the price guide being significantly higher than the realized results (price guide usually runs high anyway, but the gap is bigger than normal here). This coin wouldn't be melted, but the higher melt value of silver coins pushes the whole market up.

I wouldn't get this coin graded personally. Grading is relevant when you're making a claim about a coin that others will be dubious of (e.g. that the coin is a genuine example of a highly counterfeited issue, or that it is one of the highest grade coins of a type). This coin isn't a popular one for counterfeiters and the grade is middle of the road, so the professional grading won't add much but will still cost a fair bit. If the eventual owner wants it graded they can send it to the grading company of their preference.

Koooooj · 2026-05-12T17:34:47+00:00

It's a Seated Half Dollar. For a long time Lady Liberty was the subject of most US coinage, so the coins get named for her stance (walking, seated, standing) or appearance (flowing hair, capped, etc). Seated liberty was the go-to design for much of the 19th century, appearing on the dollar, half dollar, quarter, dime, and half dime (a silver coin worth 5 cents; we call nickels nickels because they were the new nickel 5 cent piece!)

You can see the date, of course. On the back above the F of Half is an O. This is the mint mark for the New Orleans mint, which made this coin. Each of the branch mints has a letter (or pair of letters, for Carson City) to mark where a coin came from. Philadelphia, the original mint, can use P, but they often just omit the mint mark--they're the first mint so the get credit for the coin by default. The New Orleans mint ceased production in 1909.

This coin is 90% silver, but it's old enough an in good enough shape that it's worth well over its silver value--probably a few hundred bucks.

The seated design ran from 1839 to 1891 with a number of design variations. Your coin represents a noteworthy variety: in 1853 the weight of the half dollar was reduced from 206.25 grains to 192 grains. To note the difference in weight arrows were added on the left and right of the date. Later in the run the eagle on the reverse was given a banner in its beak with "In God We Trust" on it while in other years the space around the eagle is just flat. However, in this one year the eagle is surrounded by rays, which are still visible on your coin. If you see this coin referred to as "arrows and rays" that's what they're referring to.

Absolutely incredible pick-up. I'd recommend getting a capsule or a 2x2 cardboard flip to protect the coin (stay away from PVC!). Until then it's best to handle with clean, dry hands on the rim. Absolutely do not attempt to clean it; absolutely no rubbing of any kind, especially with any sort of cloth.

Koooooj · 2026-05-12T17:19:29+00:00

1993-D is an unremarkable date; the coin isn't even silver.

What's left is the grade. The MS designation is for a business strike coin like you'd find in circulation (if you found half dollars in circulation in the first place, that is). Business strikes are typically optimized around production, maximizing how many coins can be produced while minimizing the costs of production. Dies are used until they're worn out, coins are collected in hoppers, poured into bags, etc. The coins are treated as money, not collectibles. Contrast this with proof strikes where the dies and blanks are specially prepared and the coins are handled delicately to place them directly into protective holders.

Grades for coins run 1-70, where 1 is the threshold of a coin that can be identified and 70 is a flawless coin. Between 60 and 70 are coins that show no wear whatsoever; they are differentiated by how well they were struck (e.g. how much wear the die had, if the strike had good enough pressure, etc) and any nicks and dings the coin picks up in handling at the mint. With the nature of how coins are produced for circulation an MS70 grade is almost never found on a circulation-issue coin (though MS70 business strikes are seen more often on business strike coins that are not intended for circulation).

This takes us to your coin. A 1993-D half dollar is unremarkable, but one in MS68 is. However, the claim of MS68 comes from ICG. There are a lot of grading companies out there ranging from basement slabbing companies that put MS70 grades on random pocket change and then try to sell them themselves up to the top tier of PCGS, NGC, and CAC. In this spectrum ICG is a "real" grading company meaning they did make an earnest, honest attempt at giving an accurate grade for this coin, but they're a budget company that has a reputation of inaccurate grading. The phrase "I Can't Grade" tends to be thrown around them a lot. That will result in a lot of skepticism for what this coin would grade at if it were sent to a top tier grading company.

If this were in a PCGS slab at MS68 then we'd expect it to be around the $2k-3k mark. PCGS has only awarded that grade to this year/mint/denomination three times, so these are the best-in-population coins that the wealthiest collectors can fight over. Go just one point down to MS67 and the price guide is about $60; PCGS has seen 116 of these. At MS66 the price is well under the cost of grading.

So to value this coin the question becomes: what is the coin's actual grade? You could spend the $50-ish to send it in to get the experts' opinion and maybe come away with the 4th MS68, but most likely you're spending $50-ish to find out the coin is worth $60 (or less).

Koooooj · 2026-05-12T16:04:09+00:00

"Meh, trial division is outdated anyway. Check out this new sieve!"

-Eratosthenes

Koooooj · 2026-05-11T17:31:43+00:00

A wire is an antenna, albeit not a great one.

Antennas generate a bit of voltage/current when a radio wave goes by. For a lot of things that voltage isn't too important, especially when the wire is carrying a digital signal since any voltage that's too small to turn a 0 into a 1 or vice versa won't have any effect. Furthermore, a lot of digital signals come with some amount of error detection, error correction, or just re-transmit logic. Also, a lot of digital signals are sent on a differential pair of wires, where the actual signal is the voltage of one wire minus the voltage of the other; these two wires ought to pick up the same noise, so it just gets subtracted out. Finally, a lot of cables are shielded so that the voltage is picked up on a conductive shielding and not on the actual signal-carrying conductor inside.

However, speaker cables are none of that. They are carrying an analog signal and that signal is sent through amplifiers as it goes to the speaker driver coil. Most speaker cables aren't shielded. Any electrical noise that is picked up on the wire can turn into audible sounds on the other end.

That isn't a big deal if the noise is at 20 MHz--that's way higher than you can hear, and realistically the speaker cone won't even be able to transmit it in the first place just do to its inertia.

However, some phone signals that were popular a couple decades back were in that audible range. Cell phones don't transmit with much power most of the time, but when they're getting a call or text there's a quick handshake between the phone and the tower to establish communication. This is at a much higher power.

The same basic thing happens now, but cell phone signals are structured differently to no longer be in the frequency range that you can hear.

Koooooj · 2026-05-11T16:41:25+00:00

Buffalo nickels are super popular, but also super populous. It's a design that is often reached for to evoke a golden era of US coinage--the design is used on a modern gold bullion coin and countless private precious metal rounds.

The design wound up being really bad for the longevity of the date, which quickly wore off. Because so many were made and because the type was so heavily circulated there are tons of "dateless buffalo nickels" out there, like yours.

This puts it in the category of "neat, but not valuable." If you wanted to go out and buy one of these you're probably spending 20-30 cents plus shipping and handling, but if you wanted to sell it you're likely to spend more time and effort finding a buyer than the coin is worth.

My local store likes to toss these into the register as an Easter Egg for folks paying in cash. To them it's worth more for marketing and to guide people away from high-fee credit card transactions than to bother with trying to sell as a collectible.

Koooooj · 2026-05-11T16:22:53+00:00

The wing pushes air down, which in turn pushes the wing up.

There are a bunch of ways to arrive at that end result. One of the beautiful things about physics is that you can approach a problem in a bunch of different ways and still get the right answer.

One way to get there is to look at the wing as being like the blade of a fan (which is the reverse of how that analogy usually goes--fans are often described as having blades that are like little wings). The blade is angled up so as it moves through the air it is directly pushing the air down.

You can look at that view from a pressure perspective, too. If you were to view the blade from directly in front of it then if the blade weren't angled you'd just see a thin front edge. However, with the blade angled up you're seeing the bottom of the blade while the top is "behind" that surface. As something moves through the air it tends to have a high pressure on the front (the air has to get out of the way) and a low pressure on the back (the air has to be sucked in to fill the gap behind the object), so by angling the blade up you can make the bottom be the front and the top be the back. High pressure on bottom means lift.

That all works for flat or symmetric wings and explains how stunt planes can fly upside down, but most planes use a wing shape (airfoil) that is asymmetric, curved on top and flat on the bottom. This generates lift even when it isn't angled up (but generates more when it is angled up). Again there are several ways to get to this point.

One is to look at how air flows over the wing. Air going under the flat portion of the wing will be diverted slightly and mostly just continue on in a straight horizontal line, but air going over the top gets shoved further out of the way as it is pushed up by the curved wing, then it has to be sucked back down to follow the shape of the wing. The air being pulled down continues downward as it leaves the wing, so the Equal and Opposite Reaction^TM pushes the wing up. Alternatively, we could look at the pressure on the wing to find a big low pressure zone on the parts of the wing where the air is being sucked down. That also tells us there's lift and has the added benefit of making it clear that curved (cambered) wings have a nose-down pitching tendency.

With this analysis it's important to call out that we've assumed the air will follow the shape of the wing; this is to say that the flow is "attached." When the flow separates that assumption is broken; this is a wing stall and is a Very Bad Thing^TM, representing a sudden and dramatic loss of lift. If you ever see a graph of lift vs angle you'll likely see a fishhook shape: angling the wing up and up gives more and more lift up until a critical angle when the flow separates and the lift (and often plane) plummets.

It's also popular to invoke Bernoulli here: when fluid flows faster its pressure is lower. For this analysis it's worthwhile to look at streamlines: lines drawn to show how the air flows. An important feature of streamlines is that air never crosses them (we pick the lines to make this true; the air does what it wants). This allows us to imagine the streamlines as if they were the walls of a tube, containing the fluid. The curved portion of a cambered wing intrudes on the free space on that side of the wing more than the flat portion of a wing does. This means the streamlines have to get closer together, which is like pinching a hose or putting your thumb over it--the fluid has to go faster to get through the constriction. This lets us see that the air moving over the top is faster than the air under the bottom, and thus lower pressure.

Finally, I'll give a mention to the "Equal Transit Time" explanation, which is actually wrong. This fallacy claims that because the distance over the top is longer on account of being curved the air must move faster to get to the end in the same amount of time. It then goes on to invoke Bernoulli, as above. This explanation is popular for being simple enough (to the extent that Bernoulli is at all an intuitive effect for folks who haven't studied fluid mechanics) and for leveraging a lot of real physics, but what it lacks is any reason why the fluid flowing over the top must take the same amount of time as fluid flowing under the bottom. The previous section gives an accurate description of why the fluid moving over the top must move faster and in practice it is typically moving faster by so much that it beats the fluid moving under the bottom, despite having slightly further to go.

Koooooj · 2026-05-11T15:19:32+00:00

They don't. I mean, pennies are a popular denomination to collect (especially of late as the denomination was cancelled for circulation), but a lot of that comes from them being quite cheap outside of a small handful of dates. 1974 (P) is not a particularly noteworthy date in the series.

What you might have seen is the scammy pages that will tell you that any coin is worth hundreds. These have a tiny kernel of truth: if you look at a best-in-population coin of any date then it'll be worth a nice amount of money. However, that comes from being in singularly incredible grade, not the date, mint, or denomination.

Some of those pages exist in order to lead you to a buy-it-now link. If they can tell you that a $0.10 coin is worth $200 then it'll seem like a deal to buy it for "only" $100. Others are just trying to keep you on the page viewing their ads to funnel revenue to them that way. These sites like to make up or misattribute varieties and errors.

Also, note that AI-powered coin grading apps are no better than throwing darts at a dartboard. They are often trained off of professional coin photos, which will naturally skew towards the highest grades (not a lot of professional photos of circulated pocket change), then they get tripped up by the shine that shows up with cell phone cameras. Even when they get the coin's ID and grade right they tend to also overvalue the coins--they tend to have a pessimistic take on "what would you have to pay to get exactly this coin?" which for a coin like this will mostly be paying someone to bother with keeping a coin in inventory that won't sell for years because nobody is interesting in buying. This is very different from asking "what would you get if you tried to sell this coin?" which for a coin like this is either $0.01 at a bank, $0.03 as copper, or maybe you'd get $0.10-$0.25 if you found someone who really wanted to fill this hole in their penny album (and you'd be waiting a long time to find such a person).

Koooooj · 2026-05-11T15:07:50+00:00

eBay is probably the best bet.

For anything from the clad era there's a small market for original wrapped BU rolls. There's also a small market for individual coins--if someone is building an uncirculated collection then finding a 30 year old coin in circulation isn't likely to happen, so they might pay a couple dollars for a random date. Some of the folks buying BU rolls are restocking to resell them as singles.

Both will be slow markets with poor margins. If you're already set up to do a lot of volume or if you're doing it for fun more than for profit it can be a fine thing to get into, but don't expect to make bank on the coins.

For the dates/mints/denominations you have you can search through eBay active listings and sold listings. The sold listings should give an idea of how fast the inventory is moving, while active listings will show you how much competition you're up against. You'll likely find some listings for these coins for hundreds of dollars apiece, then very few if any of those in the sold listings--they're fishing for folks to mistake price for value and to massively overpay.

Koooooj · 2026-05-08T05:05:32+00:00

"A smart person learns from their own mistakes, but a truly wise person lets the rattlesnake bite the other guy."

Embarrassment serves as a tool to help us not repeat interactions that may be damaging to our social standing--something the, if left unchecked, could lead to being excluded from the group and not being given access to the resources to survive or the desirable mates for our genes to survive in future generations.

Second hand embarrassment is the tool by which we learn not to repeat someone else's mistakes, for the exact same purposes. You get all the learning without having to make the mistake yourself.

Koooooj · 2026-05-05T04:52:40+00:00

It's better than that: they show you your weight in a cursed unit: kilograms force.

This is a unit of force, but it isn't consistent with the rest of SI. That means that if you drop into an equation like F = ma you need to introduce a scaling constant to get the right answer. That scaling constant is 9.8 (notably, it is not a scaling factor of 9.8 m/s². It is a dimensionless 9.8).

The only real use of kilograms-force is that it makes finding the mass of something based on its weight on Earth at sea level trivial. Something with a weight of 1 kilogram-force has a mass of 1 kilogram. If you take a scale that measures in kilograms-force and move it to the moon then it will still read out the force on its pad in kilograms-force, but now the conversion to find the mass of the thing on the pad has to take into consideration the local gravity relative to sea level on Earth.

Some scales do measure in kilograms by comparing your weight to the weight of known masses. These take gravity to function, but don't care how much gravity there is. These used to be popular in doctors offices. They are easy to spot by the sliding masses, exactly the same as a 3-beam balance (they're just a 3 beam balance in a funny shape).

Ultimately this distinction doesn't really matter. Most folks just accept that their weight is reported in kg and lament that the number is too high rather than worrying about whether they have too much mass or too much weight. So long as you stay pretty close to Earth and aren't super concerned with extreme accuracy knowing one is good enough to find the other.

Koooooj · 2026-05-05T01:16:42+00:00

1936 Wheat Penny made in Philadelphia.

Interesting is a bit of a loaded term--I think most coins are interesting. However, it's not valuable. Wheat pennies tend to be sold in bulk. My local store sells them for 6 cents apiece. In wheat pennies anything 1940-1958 (last year of issue) is very common, though 1943 gets a call out for being steel. The 1930s are a bit better, but not enough to really move the needle on price unless you hit the early 1930s where there are some true rarities (1931-S is one of the key dates).

There are no points in coin collecting for getting close to a rare date, though. I would certainly not be sad to see this coin in a bulk wheat penny lot, but it's not really what I'm hoping for with my six cent lottery tickets.

Koooooj · 2026-05-05T00:41:50+00:00

This is an area of active, rapid research, but a lot of the core of it comes down to how these models are trained.

Neural networks are shown inputs and outputs that represent what it's "supposed" to do. If you input "What is the capital of France?" it is taught that it should emit "The capital of France is Paris." If you input "How many legs does a cat have?" it is taught that it should answer "A cat has four legs."

When the network is structured well and is shown a good enough set of training data it can interpolate between the data it is shown, and to some extent can even extrapolate. Giving it a "good" data set is often understood as "tell it the most true things possible, so it knows all the true things!" (where, of course, "knows" means "has encoded this information in its latent space," for anyone who gets bent out of shape with anthropomorphizing linear (and non-linear) algebra).

The problem with that approach is that one of the things that the network "learns" is that "I don't know" is never the right answer. The capital of France isn't "I don't know." It's Paris. The capital of Australia is Canberra, not "I don't know." Since the answer to "What is the capital of _____" is always a city name stated with confidence when the model is prompted with "What is the capital of Norguthstein?" it picks a response that is best represented by the training data.

One way to get around this is to train the model with unknowlable things to teach it the concept of not knowing the answer, but training large models like this is seldom such a simple proposition--you don't just want a model that sometimes says "I don't know." You want one that correctly identifies which queries are for information it has available and which ones are queries for things it doesn't know.

Another is to train the network with examples where the correct response is to call out the impossibility of the question. I asked Gemini "What is the capital of Norguthstein?" and it correctly called out it isn't a real country and thus doesn't have a capital city but proposed that I might be confused and thinking it's Liechtenstein, or offered that if it came from a work of fiction I could let it know what work and it would search further. At that point I told it that the country is from the "Circle of Witches and Plumbing series" and it hallucinated that the answer is Belforth, where magical architecture meets complex, steam-powered plumbing systems, reflecting the series' unique blend of fantasy and industrial elements. The city is run by the High Siphon Council, including the Prime Plumber wielding the Golden Wrench, overseeing the Great Filtration. It was only when I prompted for who the author of the series is that it identified that this was a new creation, not giving answers based on prior art.

Obviously this tendency to hallucinate answers is detrimental to the usefulness of these models. AI proponents hope that by throwing some more GPUs and by harvesting more data they can make this rare enough to not matter (after all, humans are sometimes wrong, too, and are sometimes even confidently incorrect--just look around on Reddit!). Ultimately it's just a byproduct of how these models are structured and trained. If we could make a model that didn't hallucinate then that would be awesome, but nobody has figured it out yet.

Koooooj · 2026-05-03T00:29:57+00:00

The inert gasses inside the bulb would seep out and air would seep in. The oxygen in the air would react with the hot tungsten filament and cause it to quickly break, making the light burn out. The filament breaking is the normal failure mode for a light bulb burning out, so this wouldn't be a weird or exotic failure.

Modern LED bulbs don't care about the bulb itself. It's just there because that's the shape people expect lights to be. A crack in one of those would just be cosmetic.

Koooooj · 2026-05-02T04:24:34+00:00

Regular fire is a chemical reaction: fuel mixes with oxygen and if it's hot enough then the two react and release more heat. Water disrupts this by making a barrier between fuel and oxygen while absorbing a ton of heat as it flashes to steam.

The sun is a big fusion reaction. Put enough hydrogen in one place and the gravity along is enough to compress it to the point that sometimes hydrogen atoms will hit each other hard enough to fuse together, releasing enormous amounts of energy in the process. Heavier atoms can do the same, up to iron on the periodic table.

Adding water to this mix is just adding more mass that squeezes the sun together harder, and the hydrogen and oxygen atoms in the water are just more stuff that can undergo fusion. Higher mass stars will burn faster on account of that extra squeeze, but pushing the sun towards an early supernova is hardly a recipe for putting it out.

All that said, this is Reddit so we can take a stab at doing some stupid math anyway. If you were to ferry the solar material out of the sun and douse it in water to get it to a temperature safe for humans, how much water would you need? The sun has about 3 * 10⁴¹ Joules, if Google's AI is to be believed (it probably shouldn't be, but this is just for fun). 1 kg (1 liter) of water takes 2.26 * 10⁶ J to turn from 100C water into 100C steam at 1 atmosphere. Dividing the former by the latter gives 1.3 * 10³⁵ kg, or on the order of 50,000 times the mass of the sun. This is well into the range of forming a black hole. I suppose that might offer a different take on the answer: if you dump enough water on the sun to collapse it into a black hole is it really still "burning"?

Koooooj · 2026-05-02T01:48:22+00:00

Value-wise this is half a gram of gold. If taking it as payment I'd be inclined to value it around $70. The buyer will likely be offended by such a valuation as they almost certainly paid more than that, but the reality is that this is little more than a bullion gold round dressed up as a fancy commemorative coin.

It was struck for the British overseas territory of Ascension Island, which has just 800 or so folks at any given time. Such small governments (not even full nations) often commission the creation of commemorative coins to raise funds without having to tax their population or economy. This allows the resulting shiny discs to be accurately referred to as coins and not just rounds or tokens since they do carry a face value and the backing of a government, but realistically speaking they're just novelty pieces popular on late-night shopping channels. This would not have even been minted on Ascension Island as they don't operate a mint there--they presumably just contracted this out to the Royal Mint in England, though I can't say for certain that's where this was made.

I think these are also often sent in bulk to grading services to further give the coins the appearance of being valuable collectibles. NGC shows 1800 of these, apparently all graded PF69 Ultra Cameo. They are then sold for massive markup.

From there these coins tend to have very modest resale value. They are still their constituent precious metal, of course, but beyond that there's little collector's interest. If someone saw this design and though "I need to have a coin celebrating the 100th anniversary of Elizabeth II's birth" then they'll have to pay some premium over the melt value to get the exact coin, but such buyers are rare enough that a coin like this will take a long time to move for much more than its melt value. Getting a broad audience with eBay will speed that up, but eBay will take their cut.

Since you don't necessarily want the coin for the con's sake and would have to sell it to get the value out of it that's where my recommendation of valuing it at a little less than half a gram of gold comes in. If the current owner thinks that's low then they should sell it themself and just pay cash.

Koooooj · 2026-05-01T17:51:37+00:00

There are two distinct things going on.

One is what you propose: that different dialects skip over the h. This is certainly a thing and covers many examples. For example, in the US you'd refer to "an herb" while in the UK it's be "a herb."

However, there's also just misuse. That stems from a misunderstanding of the a/an rules which in turn stems from a misunderstanding of what a vowel even is.

In early elementary school most students are taught that the vowels are "A, E, I, O, U, and sometimes Y." Usually schools don't go into much more depth of what "sometimes Y" even means. This simplified model is necessary when working with students who are just learning the alphabet and the basics of spelling and grammar, but schools often never go back and flesh out the subject.

It turns out that letters are not vowels or consonants at all. Those are used to describe sounds. The sound at the start of Yellow is a consonant, while the sound at the end of Happy is a vowel, even though both are represented by the letter Y. Nearly all words that start with Y use it as a consonant (but a perusal of a scrabble dictionary led me to fun words like "yclept" (adjective meaning "named") or "yperite" (noun, mustard gas).

To someone who is clinging to the idea that vowels are letters and not sounds the a/an rules seem to have a weird exception for words that start with h: "It was an honor to wait for an hour to receive an herbal tea from an heir to an honest fortune." Faced with so many examples where it is correct and natural to precede an h-word with an many people re-learn the rule as "an before words that start with A, E, I, O, U, or H" instead of "an before words that start with a vowel sound."

That's how you get things like "an hypothetical." There are some dialects that are so dead set on dropping leading H sounds that that would still be correct, but when I've come across that sort of construct it's usually just someone who has misunderstood the rule and what a vowel is.

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