Something Completely Different

by Michael Cain
(submitted by request)

A recent comment thread headed off into a discussion of the attractions of games and puzzles that involve combinatorial search, like Wordle or Sudoku or Freecell. Here's an example of a combinatorial puzzle. My daughter brought this home from math class when she was in eighth grade (long ago).

On the way home from work I stopped at the corner convenience store to pick up four items. The checkout clerk did things on the register and told me "$7.11, please."
"That seems too much. How did you calculate that?" I asked.
"I multiplied the four prices together."
"Aren't you supposed to add the prices?"
"Oh, right." After a moment he said, "Still $7.11."

What were the prices of the four items?

She told me the math teacher was explaining a technique he called guess and check: guess at the answer and check to see if it's correct. She thought it was stupid and clearly expected me to think the same. She was surprised when I said, "Cool! There's a whole bunch of neat math in there!" We talked about problems where you had to choose from a set of possibilities and had to find the right combination to solve the problem. That you often needed to find a clever strategy so you could find the right combination in a reasonable amount of time. We played around with this particular problem some, but didn't guess the right answer before it got tiresome. (No one else in the class guessed the right answer either.)

Some years after that I was working at an applied research lab that did lunch-time technical talks. I was asked to do one that had some math, some entertainment value, and that most of the staff would be able to follow. My recollection of the talk about the 7-11 problem is reproduced below the fold.

Oh, and open thread, because why not?

A first observation: solving combinatorial problems tends to be hard because they get big in a hurry. The 7-11 problem is simple to state, but the number of combinations of prices is 711⁴, about 255 billion. Sometimes, though, it's possible to take simpler non-combinatorial approaches and get the answer. Suppose a, b, c, and d represent the four prices. A different way to state the problem ignoring the constraint that the answer has to be exact in dollars and cents is

minimize abs(7.11-(a+b+c+d)) + abs(7.11-(a*b*c*d))

The minimum value will be zero when the variable values satisfy the sum and product requirement. Any set of values that doesn't satisfy those requirements will have a positive value. So, haul out your handy nonlinear minimization software and give it a go. "But Mike," I hear you say, "I don't have any nonlinear minimization software handy." Chances are better than you think that you do. The Windows version of Excel has included the nonlinear minimization tool Solver for most of 30 years. Many, many years ago when I was a graduate student I spent a semester working on the software that would eventually become the nonlinear optimization part of Solver. The quality of the code I inherited was so poor it led me to spend a morning and some amount of the department's money working my way through telephone operators and secretaries until I got a particular graduate student at Case Western University on the line so I could tell him, "Set foot in Austin, Texas and you're a dead man!" and hang up. But I digress...

I have my own piece of nonlinear minimization code that I've hauled around for decades. It's relatively simple-minded but can be coerced into doing useful things if the problem is not too big. The graph below shows how it progresses from an initial guess that all the prices are $1.50. It takes a while but converges to a set of prices whose sum and product are $7.11 with an error of less than 10^-13. Unfortunately, none of the individual prices are anywhere close to dollars and whole cents.  I don't have a copy of Excel on Windows any more, but these days Libre Office has a nonlinear optimization tool that uses a different algorithm than my code (or Excel's). It behaves exactly the same way as my code on this problem: $7.11 with high precision but fails the dollar-and-cents requirement.

Things are actually worse than that.  The graph below is an estimate of the cumulative distribution function for the largest price in the solution based on 200 runs from different initial guesses using my software.  The values are spread out over a wide range.  They are spread out uniformly over a wide range.  No clues here about where to look for the dollars-and-cents answer.

Back to combinatorial search we go.

---

When I was in graduate school I was notorious for solving homework-sized problems – also known as "toy" problems – by sheer brute force with a computer. From my point of view, it was a matter of how best to spend my time. Despite the reputation, I didn't actually use brute force all that often, but there were times when spending 15 minutes to write a program and five minutes to run it was a very attractive alternative to spending two hours trying to find and apply the particular bit of cleverness the professor wanted. Also, there's a certain perverse pleasure in being able to say, "The trickery finds only one answer to the problem, but not the other equally good answers that exist."

Below is the simplest brute force approach to this problem in pseudo-code. "But Mike," I hear you say again, "I was told there would be no code." Probably by someone who also told you there would be no math. Yet here you are, reading a post that's all math. This code generates every one of the 255 billion possible combinations of the four prices and tests to see if they come close to meeting the sum and product requirement. It's necessary to test for close rather than if the conditions are met exactly because computers (generally) only do approximate arithmetic with non-integer values. The goal here is to generate a small set of candidate solutions that can be checked quickly on a calculator. This code finds the correct answer: $1.20, $1.25, $1.50, and $3.16.

for (a=0.01; a<7.12; a+=0.01):
  for (b=0.01; b<7.12; b+=0.01):
    for (c=0.01; c<7.12; c+=0.01):
      for (d=0.1; d<7.12; d+=0.01):
        sum = a + b + c + d
        if (abs(sum-7.11) > 1e-6):
          next
        prod = a * b * c * d
        if (abs(prod-7.11) > 1e-6):
          next
        print "Candidate solution", a, b, c, d

---

It wouldn't have been much of a technical talk if it ended there. Back in the day, a challenge that was frequently issued in certain academic circles was "My code can find the answers faster than your code." At one time, I had the world's fastest code for solving a particular type of network optimization problem. I didn't try to claim the title because it was basically someone else's code. The people at the Naval Postgraduate Institute at the time were good at theory but bad at coding. Simply cleaning up a copy of their code increased the speed by 20%. The brute force code above found the answer in four minutes 50 seconds. Traditionally, one describes the infrastructure used. The actual code I used for timing for this post was written in C, compiled with "gcc -O3", running on an AMD Ryzen 5 processor clocked at 2.5 GHz, timed with the standard Linux time command. To suggest how long ago I did this 7-11 talk originally, the Linux machine I had beside my desk then had to run overnight to get the answer.  And yes, the code from that talk all those years ago was still in my personal archive.

How can we speed things up? The general approach to solving combinatorial search problems faster is to eliminate as many of the combinations as possible by some sort of logic. For example, one that I won't use in this post is since every price has to be one cent or greater the largest price has to be $7.08 or smaller, not $7.11. That doesn't seem like a whole lot but it would eliminate four billion naive combinations.

One of the first things to notice is looping over the value of d is a waste of time. Once a, b, and c are set, d can be calculated directly from the sum constraint. The loop on c can be cut short once it produces a calculated value for d less than 0.01 (one cent). 0.005 is used to allow for the way the computer handles non-integer values. The number of combinations where the code gets to the multiplication test is reduced to 59 million. This code finds the correct answer in 0.102 seconds.

for (a=0.01; a<7.12; a+=0.01):
  for (b=0.01; b<7.12; b+=0.01):
    for (c=0.01; c<7.12; c+=0.01):
      d = 7.11 - (a + b + c)
      if (d < 0.005):
        last
      prod = a * b * c * d
      if (abs(prod-7.11) > 1e-6):
        next
      print "Candidate solution", a, b, c, d

---

If you could see the output, you would immediately notice the code actually finds the answer 24 times, once for each of the permutations of the four values. Another simplification is to limit the loops so only the single permutation with prices in order from smallest to largest is found, shown below. Note also that since b greater than or equal to a is now guaranteed then if a is 3.56 or greater a+b is greater than 7.11 and the overall sum requirement can't be satisfied. We can safely cut the upper bounds on the loops in half. At this point we have reduced the number of combinations considered from the original 255 billion to just over 2.5 million. This version of the code finds the solution in 0.012 seconds.

for (a=0.01; a<3.56; a+=0.01):
  for (b=a; b<3.56; b+=0.01):
    for (c=b; c<3.56; c+=0.01):
      d = 7.11 - (a + b + c)
      if (d < c - 0.005):
        last
      prod = a * b * c * d
      if (abs(prod-7.11) > 1e-6):
        next
      print "Candidate solution", a, b, c, d

---

As a general rule, computers perform operations on integers faster than on floating point (non-integer) values. We can restate the problem in terms of cents. That lacks the catchy 7-11 tagline in the original statement because the sum of the cents is 711 and the product is 711,000,000. The changes in the code necessary to find a solution in cents are shown below. A solution, if found, is no longer a mere candidate. Because the use of integers is exact, it's guaranteed to be a solution. This finds the solution in 0.006 seconds – six milliseconds. This is fast enough to ask different sorts of questions and get answers in a reasonable time. $6.44 is the smallest value between $1.00 and $10.00 that has a "four prices with the same sum and product" solution. $6.75 is the smallest value in that range that has two different solutions. There are multiple values under $10.00 that have five solutions. There are many prices in that range that don't have a solution. But is this as fast as we can go? (Hint: Betteridge's Law applies.)

for (a=1; a<356; a+=1):
  for (b=a; b<356; b+=1):
    for (c=b; c<356; c+=1):
      d = 711 - (a + b + c)
      if (d < c):
        last
      prod = a * b * c * d
      if (prod != 711000000):
        next
      print "Solution", a, b, c, d

---

In the integer version of the problem, we notice that all four prices satisfy two conditions: (1) they are integers in the range from 1 to 711, inclusive, and (2) they are factors of 711,000,000. Finding the list of those factors isn't much work. This is a much smaller list of numbers than any of the above versions use, so the number of combinations is enormously smaller. For this particular problem, there are 61 factors in the list. The number of combinations considered is reduced from the original 255 billion to less than 17,000. This is on the scale of problems that were solved by hand historically. See, for example, the orbital mechanics calculations done by Johannes Kepler's graduate students minions apprentices. The computer gets the solution in 46 microseconds.

# Calculate list of factors, values strictly increasing
n = 0
for (i=1; i<712; i++):
  result = i * (711000000 / i)
  if (result == 711000000):
    factor[n++] = i

# Search over combinations of factors
for (a=0; a<n; a++):
  for (b=a; b<n; b++):
    for (c=b; c<n; c++):
      d = 711 - (factor[a] + factor[b] + factor[c])
      if (d < factor[c]):
        last
      prod = factor[a] * factor[b] * factor[c] * d
      if (prod != 711000000):
        next
      print "Solution", factor[a], factor[b], factor[c], d

---

If we actually had to work the problem out by hand we wouldn't simply step through 17,000 combinations. We would take factorization one step farther and work out the prime factorization of 711,000,000 (and you thought learning prime factorization was a waste of time):

2⁶ × 5⁶ × 3² × 79

Then we might proceed like this:

• 79 can only be a factor in one of the four prices, Call that price a. a must be one of 79, 158, 237, 316, 395, 474, or 632.
• Suppose we guess 316 (2² × 79). Subtract that from 711 and reduce the problem to finding three prices b, c, and d that sum to 395 using the remaining prime factors (2⁴ × 3² × 5⁶).
• Six of the remaining factors are five. If those were distributed two each to b, c, and d, then each would be divisible by 25 and b+c+d would be divisible by 25. But b+c+d is 395, not divisible by 25, so the two-per-price distribution of the fives can't work. Then one of the three prices must have at least three prime factors of five, so is a multiple of 125. Assign that price to b, and there are only three possible values: 125, 250, or 375.
• For each of the three guesses there is a sub-problem with two prices. If we guess 375 for b then we are looking for c and d such that c+d is 20 and the remaining prime factors are 2⁴ × 3 × 5³. We can quickly show that there's no way to split those prime factors between two numbers so that those numbers add to 20. There is no solution to the two-price sub-problem, so there is no solution to the original problem where a is 316 and b is 375.

I've never worked through the whole thing (and am not going to do it here, much to everyone's relief, no doubt) but it can probably be done in an hour or two. The kind of logic done with the prime factors is something lots of people do pretty well – particularly people who work on math puzzles – but it's more difficult to code up for the computer compared to simply listing off several thousand combinations. There are always trade-offs, of course. If the problems are big enough, and have to be solved often enough, and fast enough, then the added programming complexity may be not just desirable but necessary.

---

Why does anyone reading this post care, aside from the possible entertainment value? Because combinatorial search and other optimization algorithms are everywhere these days.

Consider the example of a UPS truck leaving a facility in the morning. The label on every package has been scanned so the destination addresses are known. There are databases of all the roads in the area, updated with information about closures for repair and such. Calculating the best route for that truck to follow is a combinatorial problem. UPS developed a piece of software that can solve that problem quickly enough that every truck has a best route loaded into the truck's navigation device before it leaves. UPS says that it saves at least tens of millions of dollars on fuel and overtime every year by following the best routes.

The training process for deep neural-network machine-learning systems such as that underlying local favorite ChatGPT attempts to solve a complicated nonlinear optimization problem. Neural networks have been around for many years. They've become important today because computers are fast enough and algorithms are good enough to allow optimizing very large neural-network systems.

SpaceX has made launch to orbit and returning the booster for reuse look routine. Computers on the booster repeatedly solve a nonlinear optimization problem that flies to a specific precise location while staying within its fuel and maneuvering constraints. Interestingly, SpaceX does not use a general purpose solver because it wouldn't be fast enough. Their software development chain includes a system written at Stanford that takes a formal description of the specific type of problem to be solved and generates code for a custom solver that runs much faster than a general purpose solver will.

Autonomous cars, if/when we get them, will employ all of these tools. Trained neural networks to recognize the objects around the car. Combinatorial optimization guided by neural network systems to decide on local changes in trajectory. At a higher level, combinatorial optimization like the one at UPS to determine both the original route and a new route if there are necessary deviations.

Everyone ought to be interested in understanding at least a bit about the increasingly ubiquitous math and software that may kill you if it gets a sufficiently wrong answer.

Comments

[JanieM]

Perhaps we should stop portraying this sort of sociopathy as a virtue and this sort of sociopath as some sort of cultural hero?

This.

I mean, we've given up even the fig leaf of hypocrisy that pretended otherwise back in the dark ages when I was young.

[bobbyp]

I mean, we've given up even the fig leaf of hypocrisy that pretended otherwise back in the dark ages when I was young.

That is a very good article, JanieM. I am reminded of the famous "exchange" as between Fitzgerald and Hemingway.

The GOP worship of mamon goes back to their very beginnings as a political movement. Reaching full flower right after the Civil War, metastisizing into full bloom during the Gilded Age, and has been a reactionary movement aging in its own special vat of social resentment (all other classes are inferior to ours) ever since, reaching its apotheosis under Reagan and Bush and a nadir of narcissim with Trump.

[wj]

The GOP worship of mamon goes back to their very beginnings as a political movement.

The Republican Party arose (peopled mostly from ex-Whigs) over the issue of slavery. But plantations dependent on slave labor produced cheap cotton, which made Northern (mostly Republican IIRC) factory owners rich. So how is opposing slavery a "worship of mamon"?

[Michael Cain]

BTW, if you didn't already know, classified docs found in Pence's Indiana home.

I was reading something by a person who has been responsible for a considerable amount of classified information over years, who said that the President and Vice President's staffs are historically terrible about proper handling of classified documents. He seemed to believe that if you went through the boxes of documents that were carted out of those offices at the end of administrations, all of them have classified documents mixed in with the other things. Just because that's how the staffs roll.

[bobbyp]

So how is opposing slavery a "worship of mamon"?

wj,

Anti-slavery was not the only item on the early GOP's agenda.

Thanks for the reply.

[CharlesWT]

A lot of documents that aren't very sensitive get classified. It's also a way of shielding documents from FOIA requests.

[wj]

bobbyp,
Anti-slavery certainly wasn't the only item on the early GOP agenda. But it was the defining one. You could hold lots of positions on various subjects (e.g. you could be pro-business) and belong to either party. But if you were pro-slavery, the Republicans were not an option.

Also, I'm having a little difficulty reconciling (Theodore) Roosevelt' progressive reforms with the worship of mammon. Could it be that you are so irate at today's GOP, as any sane person would be, that you are projecting that onto the entire history of the GOP?

[Michael Cain]

A lot of documents that aren't very sensitive get classified. It's also a way of shielding documents from FOIA requests.

Another of the things I have read is that the Prez and Veep tend to a "If it's not important enough to classify, why are you bothering me with it?" attitude. Just to cut down on the amount of decision-making that gets up to their level.

I would probably be a terrible President. My expectation for my Cabinet-level officials would be, "Take care of it. If you're uncertain about my opinion on a matter, come ask and then take care of it. Ask too often or cross me too often and I'll replace you. Keep in mind that I didn't hire you to implement your policies, I hired you to implement mine."

[wj]

"Take care of it. If you're uncertain about my opinion on a matter, come ask and then take care of it. Ask too often or cross me too often and I'll replace you. Keep in mind that I didn't hire you to implement your policies, I hired you to implement mine."

The risk being that someone who takes you literally will be reluctant to come to you and say: "We tried your policy, but it just doesn't work in the real world." People telling the boss only what he wants to hear is a problem anyway. But it seems like this could make it worse.

[ral]

wj, well Michael did write he'd make a terrible President. :-)

I am in the fortunate position of being able to work just as a senior technical consultant and give any management advice as mere "suggestions." I would hate having to be in charge.

[wj]

I would hate having to be in charge.

Amen. As far as I can tell (and I have some first-hand experience to back me up), being in charge mainly means that you spend all your time in meetings, rather than doing anything interesting.** In theory, you can think about, and make, policy. But you've pretty much got to do that evenings, weekends, and any other bits of "your own time." That or outsource it, in which case you aren't really doing it, are you?

** I suppose there should be a caveat here to the effect that it assumes you have something resembling a sense of responsibility.

[Michael Cain]

The risk being that someone who takes you literally will be reluctant to come to you and say: "We tried your policy, but it just doesn't work in the real world."

These are senior people, none of who's ego problem is that it's too small. I like to think that I can be more subtle in person that I can be in a blog comment. Sure, bring me the numbers. Just don't let me catch you having half-assed an attempt at doing it my way, then claiming it didn't work.

Biden is kinder than I am. Someone(s) military career(s) would have been over after the Afghanistan withdrawal.

[wj]

Biden is kinder than I am. Someone(s) military career(s) would have been over after the Afghanistan withdrawal.

Given the short prep time the schedule gave them when announced, some ragged ends would be understandable. A withdrawal being rather more difficult to organize than an invasion. Still, having a withdrawal plan already in hand would have been sensible, since it was obvious that it would be happening at some point.

So yes, even allowing for the fact that it was a rush job, definitely not a shining moment for not only the guy in charge but also a couple of levels down. Plus, I would say, some serious career damage would be warranted for the guy(s) in intelligence who so badly misjudged how robust the Afghan government really was(n't).

[Girl from the North Country]

He seemed to believe that if you went through the boxes of documents that were carted out of those offices at the end of administrations, all of them have classified documents mixed in with the other things. Just because that's how the staffs roll.

I absolutely believe this. So, in fact, the defining difference now should be between ex-Presidents etc who volunteer info about it, cooperate fully etc, and those who drag their feet, pretend they declassified them, and otherwise lie about it.

[Michael Cain]

Still, having a withdrawal plan already in hand would have been sensible, since it was obvious that it would be happening at some point.

My understanding of what turned up after the fact was that the people who should have done the planning made the assumption that Biden would eventually break Trump's promise and bring more troops back because that's what the planners wanted to do. If I were in Biden's shoes, that would smack too much of the military believing the President should and will just do what she/he's told.

My own opinion is that Ukraine is turning out to be the same problem at the strategic rather than tactical level. The military has spent the last 30 years preparing -- at great expense -- for the wrong war, based on faulty intelligence and assumptions.

[JanieM]

The military has spent the last 30 years preparing -- at great expense -- for the wrong war, based on faulty intelligence and assumptions.

"preparing for the wrong war" must be the cousin of "preparing for the last war" -- an adage that was probably invented sometime between the first war in human history and the next.

I get the point about Biden's agenda vs the military's, and I agree with it so far as it goes, but I also wonder if, horrifyingly enough, we might not still need the preparations for the "wrong" war in another part of the world.

*****

In the context of skimming several dKos articles about Ukraine every day plus Adam's updates at BJ, I can't remember where I got this link, which is to a Timothy Snyder lecture in Europe from several years ago. It's a deeply fascinating look at the history of the relationships amongst Germany, Ukraine, the USSR, and Russia, focusing on Hitler and WWII.

I know someone who studied WWII in college, and he's been listening to Snyder's lectures at Yale about Ukraine and says they're amazing. I'm not big on lectures and podcasts generally, I don't take in information very well out loud. But I'm halfway through my second time through the lecture at the link because there's so much to learn in it.

[JanieM]

I meant to say that I might even have gotten the Snyder link here, and if so, apologies for the duplication. But even in that case it's worth reaffirming how good it is, and offering thanks to whoever posted it.

[ral]

Thank you JanieM for that link to Timothy Snyder's lecture! From 2017!

[Pro Bono]

"preparing for the last war" -- an adage that was probably invented sometime between the first war in human history and the next

It seems to have been first recorded in 1929 by J.L.Schley. But he doesn't say when he thinks the words were first uttered.

[wj]

I do have one question regarding Janie's first link. Prof Snyder makes a big point about how the whole point of WW II for Hitler was to colonize Ukraine. How does that square with the Molotov–Ribbentrop Pact? Not to mention the invasion of France. Why not just slam through Poland to Ukraine?

[JanieM]

wj, maybe his semester-long class has answers for you? (the other link in my comment) Maybe Hitler thought he could walk and chew gum at the same time.

[GftNC]

Just a brief OT digression. Tonight is Burns Night, which makes me think of two things.

The first (which I may have told you before, but I love it so much I can't not tell you again) is the story Stephen Fry told about a friend who had gone to a Burns Night celebration in Germany, where the ceremony was conducted in German, and on the other side of the page was the translation into English from the German, rather than using the original. In the Address to the Haggis, Burns's words:

Great chieftain o' the puddin'-race

were translated as

Mighty fuhrer of the the sausage people

And the second thing I wanted to send your way, for anybody who loves Burns's poetry, or a beautiful voice, is this album by the wonderful Eddie Reader, with some of the poems set to beautiful music. This is the third track, a great tribute to a friend You're Welcome Willie Stewart, but there are lots of gems, some bawdy but in (to me) impenetrable dialect:

https://www.youtube.com/watch?v=or29Q_FUi3M&list=OLAK5uy_lxHIQIfnL2Z_2Cqj_fqnDpSbHypyO_A5U&index=3

[GftNC]

Also, I can't resist, such a beautiful version of Ae Fond Kiss

https://www.youtube.com/watch?v=l7fi6AZQUCQ&list=OLAK5uy_lxHIQIfnL2Z_2Cqj_fqnDpSbHypyO_A5U&index=4

[wj]

wj, maybe his semester-long class has answers for you?

Working my way thru it now.

He has an amusing aside: Constitutional originalism is self contradictory. Because there is one thing that the Constitution does NOT say -- it does not say that is can only be understood/interpreted in the sense that it, and its words, were meant when it was written.

[Snarki, child of Loki]

"He has an amusing aside: Constitutional originalism is self contradictory. Because there is one thing that the Constitution does NOT say -- it does not say that is can only be understood/interpreted in the sense that it, and its words, were meant when it was written."

I think Gödel might have a theorem that is applicable here.

[Michael Cain]

I think Gödel might have a theorem that is applicable here.

I am but a humble applied mathematician. There may be true statements that are unprovable, but there are enough provable statements to be useful.

From a political and judicial perspective, I simply observe that pitchforks and torches, or the modern equivalents, provably exist.

[JanieM]

Speaking of pitchforks and torches, Snyder's wiki page says this (I am actually starting to hope his later prophecies are not as spot on as his earlier ones):

In a May 2017 interview with Salon, he warned that the Trump administration would attempt to subvert democracy by declaring a state of emergency and take full control of the government, similar to Hitler's Reichstag fire: "it's pretty much inevitable that they will try."[39] According to Snyder, "Trump's campaign for president of the United States was basically a Russian operation." Snyder also warned that Trump's lies would lead to tyranny.[40]

In January 2021, Snyder published a New York Times essay on the future of the GOP in response to the siege of the United States Capitol, blaming Trump and his "enablers", Senators Ted Cruz and Josh Hawley, for the insurrection fueled by their claims of election fraud, writing that "the breakers have an even stronger reason to see Trump disappear: It is impossible to inherit from someone who is still around. Seizing Trump's big lie might appear to be a gesture of support. In fact it expresses a wish for his political death."[41]

[ral]

For those who recall Fafblog,

I simply observe that pitchforks and torches, or the modern equivalents, provably exist.

According to your limited perception.

[Snarki, child of Loki]

"For those who recall Fafblog"
..it was the BEST blog.

[wj]

I simply observe that pitchforks and torches, or the modern equivalents, provably exist.

Who you gonna believe, me or your lying eyes?

Remember the days when you could say "I'm from Missouri. That means show me."? With the strong implicit implication that, if you did show him, he would believe. But, no more -- witness Sen Hawley.

[nous]

I've been following this story for a while, and this latest wrinkle in the story seems like a very fraught one which I think Sturgeon is handling as well as she can.

https://www.theguardian.com/uk-news/2023/jan/26/trans-woman-isla-bryson-found-guilty-rape-not-be-held-in-womens-prison-sturgeon

I'd add as part of all this that I think there should probably be special facilities of some sort for criminals with a history of sexual violence, and that all carceral spaces should have safeguards in place to protect incarcerated people of any gender from sexual violence.

Doesn't seem like it should be treated primarily as a gender issue.

Also, I think that Sunak's decision to block the Scottish Parliament's gender recognition bill are going to create serious backlash and push Scotland further towards breaking away.

[wj]

all carceral spaces should have safeguards in place to protect incarcerated people of any gender from sexual violence.

From what I have (casually!) come across, that's going to take some major reforms to implement. Rape, after all, isn't generally about sex so much as power and control. And prison populations are big on that.

[wj]

Talk about a class act!
Arizona Republicans exempt lawmakers from the state’s open-records law

One can understand their motivation. Those emails from 2020, when they were jamming thru demands for a recount, were just too, too embarrassing.

[nous]

From what I have (casually!) come across, that's going to take some major reforms to implement.

Agreed. Make it so.