Lessons learned

Here are two problems I encountered while teaching problem-solving:

1). My students aren't attuned to what I'm trying to teach. When they see a problem, they're immediately filtering for "what are the steps for this problem type?" There's a lot of resistance to the idea that there is no algorithm to learn, and even more to the idea that we won't look at a single "type" of problem over and over again; instead each problem will look totally new every time.

2). Even when they get better at problem-solving, they feel adrift because there's nothing concrete they can point to and say "THIS is what I learned". As a result, they feel anxious and unsatisfied, because it's hard for them to know what they're doing or if they're getting better.

To address these problems, I'd like to give students a general algorithm for problem-solving. This will give them steps to follow when facing a new problem, but also be what I want them learn. I can introduce it early and have it appear (to start) on quizes as pure regurgitation. The goal is for the problem-solving steps to be so internalized as to become habits of mind (eventually). I can introduce heuristics a few at a time, and ideally have some accountability structure that will make students feel like they're making progress towards improvement on those heuristics.

Hopefully this will give them a cognitive reference point so they feel (and can explain to others) exactly what they're getting better at.

Hypothetical v. Counterfactual

Jen told me a story about an episode of 30 days she saw in which an athiest lived with a fundamentalist Christian. The atheist was trying to explain why she was uncomfortable with the phrase "In God we Trust" on our currency. The Christian kept saying things like, "Well, that's normal; that's just what our money says, what's the problem?" She attempted to explain using a counter-factual: "Imagine if all our money said: There is no God. How would that make you feel?" Apparently the man seemed unable (or maybe unwilling) to consider this possibility. He just kept repeating things like: "But that's not what our money says."

Of course, this misses the whole point. But I have definitely had my share of this same experience in all sorts of situations. I ask someone to consider a scenrio to illustrate a point, and often get the reply back: "but things aren't that way" or "but things would never be like that".

This initially confused me; people should be good at this kind of reasoning since we all have to use hypothetical reasoning all the time. We imagine what if we did this or that, imagine the consequences and use that to guide our actions. So what's the difference?

My hypothesis is that the difference may lie in how well someone is able to imagine the alternative world. Hypothetical situations (what if I did X) may be generally easy for people to imagine. Counter-factuals that involve more drastic changes in the world (what if we lost WWII; what if there were a God? What if there were no God?) are, for some reason, harder for some people to imagine.

It's difficult for me to separate ability from inclination here; are they really unable to imagine a counter-factual situation in detail, or just unwilling to expend the effort to imagine it (they don't see the point, maybe)? My gut feeling is that usually the scenario they're being asked to imagine (in an argument, say) is both difficult and _unpleasant_ to imagine. These may be enough to make most people resist the attempt ("but it wouldn't be like that"), rather than really try and grasp the point.

People I've discussed this with in the past often identify this as a failure of logic or rationality--people are too dumb to get it. I now think it's a failure of imagination and of trust--a difficulty and discomfort with imagining things one finds unpleasant.

If we were going to rectify this kind of failure in school, the solution probably isn't a focus on the logical structure of such arguments. It probably will have more to do with creating safe environments to use, experience, and appreciate the legitimacy of the technique.

"What's the next step?" v. "What do I need?"

Today I had an interesting experience that revealed a large gap between how my students and I think about the problems we've been doing. This difference corresponds to how you learn the algorithm for a specific skill vs how you tackle a problem you've never seen before.

If you're learning a specific skill (say, how to solve a quadratic equation), the process generally involves recognizing some cue that tells you what to do next, over and over again. The operative question is "What's the next step?"

If you have to solve a problem you've never seen before, you can't ask yourself "What's the next step?", because the whole point is that you don't know; maybe no one knows. So instead you have to ask yourself "What do I want to find out?", "What do I know already?", and "How can I start to connect the two?"

It's astonishing how few of the problems students get in school require them to ask themselves these three questions. Even if a problem starts out requiring these genuine problem solving questions, it's extremely easy for it to become "proceduralized"--either when I (or other students) give too many hints and reduce it to following-the-steps, or if we do the problem so many times that it becomes a "type" of problem the students recognize "the steps" for.

Students (usually) do a pretty good job listing what they know, and what they want to figure out; the whole trick is how to connect them together. You could "work forwards" by asking what else you can figure out with what you know. It might not seem directly relevant, but if you keep figuring out more and more things, eventually you may see a connection to what you want to know. You can also work backwards by identifying what key piece of information would let you solve your problem. This sets a new sub-goal to aim at.

My students seem to get this idea in general, but have trouble applying it. The specific phrasing of the questions I ask seem to make a big difference. To me, all these questions are equivalent:

• What could you figure out from here?
• What could you try to do next?
• What could you try to figure out next?
• What facts could you try and figure out about this situation?
• What information could you try and figure out next?
  

In class, some of these phrasings would get blank looks, and others immediate responses. In some cases, the problem was using too abstract a word ("information") that couldn't serve the guiding purpose a more concrete word ("facts") could. In other cases, I think some students think more naturally in terms of goals (things to find out), and other in terms of actions or operations (things to try).

For struggling students, it was difficult to get them to think of more than 1 thing they could try. I'd given them a diagram with two perpendicular lines, the equation of one of the lines, and a point on the other. I was hoping they'd suggest things to try such as finding the equation of the other line, finding the point of intersection, finding the x or y intercepts of either line, etc. None of these seemed to present themselves as salient pieces of information. I tried to prompt them by asking what kinds of questions they'd been asked before about situations like this, or what skills they remembered from algebra I or II about lines, points, or perpendicular lines, but without specific queues to trigger their memory, they weren't able to brain-storm a list effectively. At least not individually.

For my last block of the day we started as an entire class. I wrote the diagram on the board, and started a list with the two pieces of information we had (the equation of one line, and the coordinates of the point on the other), as well as the piece of information we were trying to find in the problem. Then we played a game where I would hand the pen to a student who would silently add a new "fact" about the diagram to our list of facts to figure out. They could add labels or additional points or lines to the diagram if they wanted. If they were really stuck, they could hand the pen to another student.

This seemed to work. Students were able to get a good list, extending or modifying ideas that were already in the list. It was interesting to notice that sometimes students would add a potentially relevant addition to the drawing (such as a line that made a promising-looking right triangle with existing lines), while a few would add a seemingly random line or point just for the sake of extending the list.

We had a discussion afterwards about which facts seemed more or less "helpful" in solving the problem. Again, as a class we were able to formulate a series of sub-goals to lead from the givens to the solution.

I think this general approach could work for small group problem solving. The main obstacle, I think, is actually getting the students to follow the process. As a whole class, they were happy to do it, but in small groups they tend to focus directly in on the problem, and seem to regard this sort of activity as long and circuitous.

Some ideas I'd like to try are these:
* Give them a situation and tell them to list, and find, as many things about the situation as possible (so there is no specific "problem" to solve).

* Scaffold the process by providing some of the steps. E.g. on the first problem, I provide the list of facts and they figure out how they connect together. Next time, we generate the list together first. After that, I prompt them to do it on their own. After that, I just give them the problem with no prompting to think about all the things they might try.

* Have the work on half-sized poster paper. Each sub-goal they formulate goes on a 3x5 card. They're reponsible (as part of their graded work) for creating and sequencing the cards on the paper (to show their problem-solving plan). After they have their plan, they can actually show the calculations on the paper.

With respect to how each student understands the specific self-questions involved in working forwards or working backwards, I think it might be best to have students say the idea in their own words, and we can just create a small collection of student phrasings. In my prior experience, this works much better than me trying to figure out the clearest way to express it, since student phrasings tend to make sense to other students. Having a diversity of phrasings will also increase the probability that one of them will catch.

Anyway, a lot of interesting issues to think about. More updates on the teaching of basic problem-solving later!

Rationalization

On NPR just now there's a novelist discussing his new novel. The main character is a reporter, reporting on "sex tours", who intends to remain detached, but ends up participating in the tours himself.

Terry Gross reads several excerpts from the novel of the different justifications that the men on the tour give for their participation. One considers how many different services are considered legitimate: we pay people to carry our things, clean our houses, massage our bodies; so why the special standard for sex? (presumably the gentleman in question thinks there isn't an important difference). Another considers the argument that it's dangerous for women and reasons that there are lots of risky jobs in society: is being a sex worker riskier than being a police officer or fireman?

At this point the novelist said something that set me off; he said that this illustrates the problem with "words". He claimed that your moral sense has to be deeper than, beyond "words", because words can lead you astray (as they did his main character).

Let's re-phrase what he really seems to be saying: Often our reasons (those "words") are really rationalizations, and we can too-easily convince ourselves with spurious arguments. If your "moral compass" is immune to such reasoning, you can escape the danger of rationalizing.

This is perfectly true, but neglects the equally compelling danger of the alternative: having a moral sense that isn't sensitive to reason means you can never check whether your moral compass makes any sense. If rationalizing is pretending you don't see, dogmatism is never looking in the first place.

I don't want to overstate people's (myself included) ability to correctly reason--I think all evidence is that it's pretty poor. But I think it's the only weapon we've got. The danger of rationalizing isn't that we've done too much reasoning; the problem is that we haven't done enough. Rationalizations, if such they are, shouldn't be able to withstand closer scrutiny. The problem is that we stop as soon as we get to an answer we like and don't look at them as closely as we really should.

I don't see many good practical prospects for improving our situation. How do you compel people to consider more deeply? The problem is even worse than it may seem, since there are a lot of people who admittedly continue to do things they've judged that they shouldn't. These, perhaps, are the more honest among us who recognize they won't change, but at least don't try to rationalize their choices. What's to be done?

I have no idea, but I do know that a promising way to deal with individual weakness is through the social bonds formed in groups; there are plenty of "keep-each-other-strong" organizations in other areas; so why not these? Religious groups are particularly well-positioned to do this since they already have the infrastructure in place, so to speak.

The foundation of such a group would be a collective recognition that no one probably knows the truth about things, and our best shot at figuring anything out is through challenging, though respectful, dialogue and that everyone needs help to become who they wish they were.

argument and propaganda

Jen had posed two questions in an earlier post:

How is argument different from force, and why is it preferable?

Of course, argument can be seen as a kind of force; after all, you're trying to logically (instead of physically) compel someone to accept your position. But we think of argument as a legitimate means to convince someone, whereas pure rhetoric or propaganda represent an illegitimate means of "force" through psychological coersian. So, to re-phrase the question, what's the difference between convincing and coercing? What makes one legitimate and the other not?

The major difference between these has to do with the use of reasons. As a first pass, we could say that argument provides reasons, whereas propaganda attempts to bypass them. Ridicule (name-calling, stereotyping), for example, can be a way to get someone to dismiss a position without ever considering reasons for or against it. More generally, propaganda often functions by trying to associate a position directly with something that will be evaluated positively or negatively, in the hopes that these feelings will transfer.

This isn't quite sufficient, though. There are other forms of psychological coersian which don't it this model. Lying, for example, involves providing false information in the hopes that it will lead someone else's reasoning in the direction you want. Deliberate oversimplification would be another example; it also aims to influence the other person by using their reason.

I think the real unifying feature of these illegitimate techniques is their attempt to manipulate. In this sense, legitimate argument is an attempt to work with (cooperate with) someone's capacity to reason. The alternatives try to "work against" it. Let's flesh this out a bit more.

Providing reasons isn't the distinctive feature of legitimate argument, since the alternatives do that, too. Legitimate argument involves helping someone else's reasoning system do for itself what they would wish it to be able to do anyway. We do this, for example, by providing accurate and relevant information, or by pointing out logical inferences. It's very much like helping someone do a math problem. You can point out things they didn't see, but would have wished they could see.

This analogy works because both activities involve trying to find the truth about something. Because arguments are so often framed as a debate, it's easy to think the purpose of argument is to convince someone. The real purpose of legitimate argument is to help them see the same truth you do.

Let's consider some difficulties:

#1 "What if their reasoning system is so faulty (or, "different", let's say) that "helping it do what they would want to do with it anyway" violates your own standards of reason?" For example, how do you legitimately convince a fundamentalist Christian who only believes in the literal word of scripture that evolution is right?

One option here would be to accept that sometimes you just can't get what you want with legitimate argument. I think this is dangerous. For the above case, I would seek principals of reason you both do accept and leverage those. It will probably be a long (maybe life-long) conversation, because before you can tackle evolution, you might need to convince them (based on shared principals of reason) to modify how they reason.

Those who are severely mental abnormal are a much more difficult example. Patients whose perception of the world, or faculties of reasoning are radically different may, in the end, be unreachable by a shared process of reasoning. Just because someone's reasoning process is different, however, doesn't mean that it shares no commonalities with your own, or that you can't build commonalities.

#2: "Your characterization implies that if there is no external truth, there is no possibility for legitimate argument, since such argument involves helping them come to see this truth." For example, when discussing issues of personal taste, aesthetics, and arguably, morality, the "truth" might just be a certain way of looking at the world; and no one way is fundamentally better than any other.

These are interesting cases. Let's consider something definitely subjective, such as whether "chocolate is more delicious than vanilla." It's definitely wrong to say that a legitimate argument for this position involves helping the other person discover the truth about this statement. But a legitimate argument for this claim might involve helping someone see if the truth _for them_ is the same as the truth _for you_. You do this by describing aspects of your own experience that are decisive in the hopes that they may discover they agree. This need not be manipulative.

This same idea seems as if it extends to other aesthetic questions fairly easily. What about moral questions?

This idea is similar to that of "framing" an issue. Linguists have long thought that the specific words you choose to describe something carry with them a set of assumptions that will partially determine what makes "common sense". George Lakoff discusses the use of framing for political/moral questions in his books _Moral Politics_ and _Don't Think of an Elephant_.

At first glance, framing can look a lot like illegitimate manipulation. After all, the words you choose can establish "hidden" assumptions that will influence the other person's reasoning system.

Certainly, frames can (and are) used manipulatively in this way. The question is, can they be used as part of legitimate argument? We all frame issues all the time anyway; we can't help it--it's the way our brains and language work.

So if frames are always manipulative, this might put a serious kink in any hopes of purity.

I said that a frame carries with it certain assumptions. What are these assumptions? They represent a particular way of understanding a situation; a way of looking at the world. In fact, they underlie how we experience the world. I think they also contain a mix of objective and subjective elements.

This makes them difficult to know what to do with.

If frames were purely my subjective way of looking at things, then using a frame might be like testing out to see if someone else also finds that way of looking at the world natural. It would be directly analogous to the painting case.

Too often, however, the assumptions a frame brings have elements that could be checked objectively. The fact that a frame hides these, however, means that typically they are not.

I will leave the issue of framing for consideration in another post.

Small Rant

It has apparently been widely reported in conservative (and other) media that in a focus group conducted by Fox news, 25 Obama supporters couldn't name a single one of his legislative accomplishments. I heard a group of people on the train today recount this fact (“discuss” would be too strong a word), shaking their heads and clucking their tongues: “...You would think that his supporters would know something about him.” And thus, the subject was dismissed; which, of course, was the point of the “news” story.

Of course, it would be easy to perform the same stunt with any candidate. I would have bet money that I could have found 25 McCain supports on the train who wouldn't have been able to name any of his legislative accomplishments; I doubt that's the way most Americans decide who to support. Indeed, I would be surprised if the certainty of the McCain supporters in question was based on their extensive legislative knowledge.

How big a factor should it be? Elections are about the future. Knowledge of a candidates' past accomplishments (and failures) will be valuable to the extent that they can inform your interpretation of the candidate's current self-presentation and plans. As such, their utility will vary greatly; but in the end I view their role as a supportive (almost secondary) one; as one of confirmation or correction of what should be primary which is what the candidate is presenting now.

But, as I alluded to above, these considerations give too much credit to a media stunt which doesn't even appear to try addressing such worthwhile questions.

For me, politics is about two things, neither of which (in my experience) come naturally to most people: hard research, and compromise. Doing a lot of hard thought and research is the only way to know what you should think about the difficult questions of politics (and they're pretty much all difficult). Compromise is the only way you make progress towards solving them.

If most people thought about politics this way, I think we would see more humility, curiosity, uncertainty, openness and goodwill. Take social security, for example. Most people are not economists, know none of the statistics that might be relevant, and haven't really thought very carefully about ideas like "personal responsibility" they're so ready to invoke; nonetheless, many people will confidently state a position, and caricature or ridicule those whose positions differ.

I have no idea what to think about social security; it's a question I haven't thought about yet, and which, quite frankly, seems pretty daunting. Far from being ready to dig in my heals, I'm actively searching for people who can make sense of their (or any!) position for me. Sadly, most people's views seem to bottom-out in something they heard someone say on tv that they didn't think about very much.

When push comes to shove, I don't think most people are really interested in doing the work required to construct a reasoned view for themselves. I don't blame them; it's a lot of work. But their certainty and demeanor is wholly inappropriate to their understanding. It's like having strong feelings about different interpretations of quantum mechanics that you're ready to vigorously defend, even though, when it comes right down to it, you don't really know very much about physics.

Despite this, I'm amazed at what people do to avoid conceding a point to "the other side." Compromise is often seen as (if not openly declared to be) weakness. The attitude I sense from most people is that politics isn't about finding compromises we can agree on, but about winning enough power to impose your ideas on everyone else. I remember in 2004 when an NPR correspondent asked a member of a republican think-tank whether Bush had any responsibility to the democratic 49% of the country, considering the narrow margin of his victory and the fact that his party now controlled both the house, the senate and the executive branch. His answer was along the lines of: "We won. Why on earth would we give-up power to the losers?"

This sort of attitude deepens divisions, clouds clear thinking, and, in my opinion, impedes the possibility of long-term progress. What kind of progress is it if it's just reversed 4 or 8 years later?

If politics should be about scholarship and compromise, what's gone wrong? I'm not totally sure, but the media seems to be fanning the flames. The way most stations cover politics differs little from the way they cover sports. It's about who's ahead, who's strong and who's weak, what strategies or tactics are successful. Politics is too difficult and important to be trivialized in this way.

Here's my advice for discussions of politics. Try to ask more questions than you answer. Try to figure out how your opponents' view makes sense to them. Find something you can agree with before finding something to disagree about. These dictates are harder to practice than preech, and I'd be the first to admit I'm not very good at them. But they're what I aspire to and would appreciate being reminded of as frequently as possible.

This might be too hard for me, pt. 1

Here is a problem we've been working on (originally taken from wu::riddles). Here is one way to describe it:

Draw a 1x1 square. In a minute, I'm going to take a ruler and draw a straight line that will "cut" accross some part of the square. It might be through the middle, or it might only be a small section of the corner; you don't know. You should draw anything you want inside the square so you can be certain that any line I draw must cross one of the lines in your drawing. The challenge is to do this with the shortest possible amount of drawing.

The first, and most obvious, suggestion would be this:

The blue lines are the solution. Obviously, it blocks all possible red lines (3 possible red lines shown), and the fact that it's symmetric makes it look pretty short. But how could we prove it was the shortest? In particular, we wanted to be sure there wasn't some weird scatter of short lines (with a shorter total length) that would also block all possible red lines.


(Here's an imagined weird scatter with few blocked red lines. Of course, a scatter like this could consist of a huge number of almost point-like line segments....so, pretty hard to think about on a case-by-case basis....)

One friend of ours suggested the following proof (actually "proof" since it turned out to be wrong). It's an interesting and useful form of argument, however.

Idea #1: Find the shortest solution for only some of the lines

Any solution that blocks all possible lines will at least have to block all the diagonal lines (in both directions).
If we can find the shortest solution that blocks just these lines, we'll know that a full solution has to be at least that long.

Let's divide the square into a bunch of parallel "slices"...

...and find the shortest solution for each "slice" that blocks only the diagonal lines from one side to the other (direction shown as the green arrow).

In this diagram, we have a slice (in black). We're only interested in blocking the green lines parallel to the one shown. The shortest solution for this slice will be a line perpendicular to the green ones we're blocking (such as the blue line shown).

So, if that blue line is the shortest solution for a slice, we can put that same solution together for all the slices...
As you can see, this blocks all the parallel diagonal lines in that direction (though, of course, it would let other lines through). But, if we're right so far, that's the shortest possible way to block those parallel lines (since for each little slice it was the shortest).

If we also want to block the diagonal lines in the other direction, we just repeat the procedure and overlay the results:
This is a scatter that we've constructed only to block the diagonal lines. BUT, we know that, for each direction, it does so with the shortest amount of drawing.

Since an actual solution (that blocks all the lines) will have to block those diagonal ones, and we know it has to be at least as long as this scatter to block the diagonal ones, the full solution will also have to be at least as long as this scatter.

But our "x" solution is exactly that long. It's easy to see why: instead of putting each little blue segment in a random spot, we line them all up along the diagonals.

You can imagine moving each bit perpendicular to its direction, and they line-up to make the x!

So, the "x" is the shortest possible way to block all the diagonals, but it also blocks all the other lines lines! Therefore, it must be the shortest full solution, because if there was another, shorter solution, we'd know it couldn't be blocking all the diagonal lines.

Since we thought the "x" solution was probably the best, we weren't very critical of the "proof" and thought we'd done a pretty good job!

Here's the problem:

This drawing is a shorter solution.

What went wrong with the original proof? Maybe lots of things, but one clear problem was how we broke the problem down into two smaller problems, and then re-combined them.

First we asked "what's the shortest way to block all the diagonal lines in one direction? Ok, let's keep that. Now let's do the same thing in the other direction to block all the other diagonal lines."

The problem is that one little bit of our "x" solution blocks lots of one type of diagonal (green), but none of the other type of diagonal (red)...
Whereas, a little piece of our 2nd solution can block both directions at once...

So maybe, even though our small pieces aren't the shortest for blocking lines in ONE direction or the other, they're the shortest for blocking blocking them together. Their overall blocking efficiency is better (we use this idea more in idea #3 below).

Idea #2: A Connected Solution is Always Shortest

The solution which beat the "x" is formed by adding two imaginary points so that, when you connect all the dots, they form 120-degree angles.

Someone pointed out that there's a theorem which says that this same method will produce the shortest path connecting any number of dots in the plane. So that seemed promising.

Of course, a shortest path connecting points is different than a shortest drawing that will block all possible lines, but it still seemed like a step in the right direction. Maybe we could connect the two problems together...

I hoped that this solution was, in fact, the shortest and we could prove it with a two-step argument:

1). If there's a "disconnected solution" (like a scatter), there will be a "connected solution" (continuous lines with no breaks) that's the same length or shorter.

2). Any "connected solution" must connect the vertecies of the square with each other.

3). Therefore, because of the shortest-path theorem, our solution will also be the shortest blocking solution.

However, we didn't get very far along this path until we found...

This was the shortest solution yet. It also showed the my old hope was wrong. I had hoped that the shortest solution would always be a connected one. Now we know that can't possibly be true.

Curious Student: "But wait! Isn't it possible that you could find an even shorter solution that is connected? Then you'd be right that the shortest solution is a connected one!"

Sadly, no. That theorem about the shortest connecting path implies that the shortest connected solution is the one we found. This disconnected solution is shorter. So if we find an even shorter solution than this, we know it won't be a connected one; it will have breaks in it.

Idea #3: Try A Smaller Problem. Measure Blocking Efficiency.

Since we didn't have a good way to analyze the square, we decided to try the simplest case we could think of: the equilateral triangle.

Here is the shortest solution we found for the equilateral triangle:

Once again we have the problem of how to prove this is the shortest.

Nick was hoping to use an idea similar to the last proof that didn't work. Here's the idea:

Any solution that blocks all lines will at least have to block all lines perpendicular to the three sides of the triangle.

If I have a random tiny line segment, I can measure how much of this blocking it does by measuring how long a "shadow" it would cast on each side of the triangle if I shined a light behind it (perpendicular to the wall) (and add them up).

Using these ideas, we hoped that we could show that our solution is the shortest by showing that it's not possible to have a random small piece that blocks more than a random small piece (of the same length) from our solution.

In other words, the orientation of the lines in our solution gives you the maximum amount of blocking possible. That would show that it was the shortest solution for lines perpendicular to the 3 sides. Since our solution also blocks the other lines, it would be the shortest overall solution (by an argument similar to our idea #1).

Unfortunately, we found that a small piece (in isolation) blocks the most when it's parallel to one of the sides; the pieces in our solution are all perpendicular.

This gave me a new idea, however. How much blocking a solution does depends on two things: how much blocking each little piece does, and whether or not any pieces are blocking the same lines (how much redundant blocking there is).

If you want to make your solution shorter, you could either try to change the orientation of your lines so that they block more, or you could move them around so that their blocking isn't redundant.

This led us to...

Idea #5: Measure Redundant Blocking

Here was my idea...

1). Find a measure of how much total blocking needs to be done.
2). Measure the maximum amount of blocking a given tiny line segment can do
3). This will let us calculate a theoretical shortest solution solution. If we assume no redundant blocking, the shortest solution will be...

(total blocking needed) / (how much blocking a unit of solution can do) = total units of length in solution.

4). Calculate the amount of redundant blocking in our proposed shortest solution.
5). Show (somehow) that exactly that much redundancy is required for any 100% blocking solution.

The really hard part of this is step 5.

But steps 1 and 2 are also tricky because they require us to measure blocking in some way. Before we were only measure how many of certain types of lines were being blocked. I wanted a way to measure all the lines being blocked.

How can we do it? Here was our idea...

...inscribe our shape in a circle. Now, the lines you need to block can be seen as all the lines going from one arc of the circle to another.

Here's a picture of what I mean. The three arcs are in different colors. Any line starting in one arc and ending in another is a line we need to block.

This gives us a possible way to measure how much blocking a little line segment does: Imagine a light on the circle, shining onto the segment. Measure the length of the "shadow" it casts on the other side. This is a measure of how many lines it blocks from that point.

To find out how many total lines it blocks, imagine slowly rotating that flashlight around the circle and adding up all the arc lengths. When you've gone all the way around, this is the blocking measure of this segment.

Mathematically speaking, if we have a function to find the arc length, and integrate it from 0 to 360 (as the light source moves around the circle), this should allow us to measure it. We haven't made this integral yet because it seemed hard and we're hoping if we're clever enough we won't have to.

My great hope was that the total amount of blocking of a segment only depends on its length; not on its angle or location within the triangle. The reason this would be good is that a certain length of solution would directly yield a certain total length of blocking. Then, to find the shortest solution, all we have to do is find an arrangement which minimizes redundant blocking. So, instead of minimizing length of solution, we'd be minimizing redundancy of solution.

Unfortunately (for reasons I won't go into), it doesn't seem like that's going to work, though I'm not 100% convinced yet.

That's all I'm going to write for now....perhaps more later...

But first:

What's making this problem hard?

The difficulty we've been having in this problem is that we can't break it down into simple sub-problems we can solve independently.

A short solution depends on two things: the "blocking efficiency" of each little piece, and the amount of redundant blocking for the arrangement. The blocking efficiency seems to depend on the angle and location of each little piece; and the "blocking" of the pieces need to be balanced between all the directions that require blocking. The amount of blocking depends on how each piece is oriented with respect to every other piece; it's a global property of the solution. And so far we can't think of an easy way to divide all the possibilities into cases we can treat separately.

If anyone has new ideas, we'd love to hear them!