1 {photo of painted carousel horses}
1 Caption: Are these horses?

Leaves make food by photosynthesis.

Douglas Hofstadter has championed a theory of how the human mind works which is based on analogies. It comes out in his book Fluid Concepts and Creative Analogies. His argument is (put very simply; I am not really doing it justice here) that the world is a complex place and there are many, many things that are too complicated to understand on a deep and fundamental level. Take for one example: photosynthesis. How do our minds understand photosynthesis?

You might think: "We'll, it's a process whereby chloroplasts in plant cells take in energy from sunlight and carbon dioxide and water and produce sugars and oxygen." This is your brain parroting a piece of information that it has learnt to associate with the word "photosynthesis". But what does it actually mean? How, exactly, is this process accomplished? What are the details of the chemical and energetic pathways, what is the underlying quantum mechanics of what is going on? Why does all of that stuff happen in the way it does?

If you're a physical biochemist specialising in plant metabolism, you have a better idea than most people, but most people aren't and their knowledge stops short at an earlier stage. Yet most of you reading this are comfortable that you know something about photosynthesis and "how it works". What for you is filling in that gap of understanding between "plant makes sugar and oxygen" and being able to write down and understand the equations governing the interaction of every subatomic particle involved in that process? The answer, Hofstadter says, is analogy.

Analogies bridge gaps... like a bridge.

To avoid this sort of gaping hole in our understanding of the world, our brains build analogies to bridge the gaps. A plant takes carbon dioxide and water, and with a bit of energy turns them into sugar and oxygen. Without digging deeper into the physical chemistry, we can understand this as a process that resembles other processes with which we are more familiar. In a factory, for example, a production line might take two materials, add some energy in the form of workers (or industrial robots) who can do work on those materials, and produce a new object and some leftover scrap. We then draw analogy lines from the input materials to carbon dioxide and water, from the new object to sugar, and from the scrap material to oxygen. Armed with this analogy in our minds, we now feel that we understand photosynthesis.

We may not do this analogising explicitly, but Hofstadter argues that our brains do it implicitly. Or brain fills in holes in our understanding with analogous concepts, so that we're comfortable that we understand the entire procedure. If our brain can find a workable analogy, all is well. If the thing we're trying to understand is so abstruse that we can't find a good analogy, then we struggle to understand it. Our ongoing education is a process of taking crude analogies we learnt when we were younger, and replacing them with better information and finer-grained analogies.

Perhaps you may not agree with Hofstadter's thesis, but to me it sounds like a feasible explanation for at least some of the process of learning and understanding. I know that when I try to learn something, I seek out similarities to things I'm already familiar with to help it all come together in my mind and be something I can recall later.

Peanut butter. And something analogous.

It's also very similar to how we teach children about the world. When a child wants to know how something works, and we know the full explanation (at least as far as we understand it ourselves) would be too much detail right now, we draw analogies for them. "How does a plant grow?" "Well, it takes sunlight and water and air and makes food for itself." The impression the child gets might be something a bit like a person taking bread and peanut butter and making a sandwich, then eating the sandwich to grow big and strong, and that may well be good enough for them at this point in time. This is also how children are taught many things in our schools. There the analogies are polished and honed by decades of teaching experience and recorded in textbooks. As the ages of the students increase, the analogies get more sophisticated and deeper in their explanatory power.

Analogies are a big part of what I try to do in these very annotations. I try to make topics of varying difficulty somewhat easier to understand. I try to help your brains draw a web of connecting concepts and analogies that make use of what you already know, in order to understand new things - things that can be thought of as having some similarities which perhaps you never saw before. Connecting to last week's topic of memory, I find these sorts of conceptual cross-links extremely valuable in cementing understanding and recall in my mind. So while Hofstadter's idea may not be the be-all and end-all of learning, I find it interesting and enlightening.

Caution: surrounding areas may be slippery.

Different people make any given analogy more easily or less easily. Or minds have a certain level of rigidity or plasticity about such things. Hofstadter claims that people who are looser, or more "slippery", with their mental analogy making are better equipped to learn things. He equates it with a component of intelligence - certainly not the entirety of the concept we call "intelligence", but a significant part of it.

I was reminded of all of this stuff about analogy making, and how it is more natural or less natural for different people, by events around the recent CiSRA Puzzle Competition, which a group of friends and I organise. The sort of puzzles we create require various levels of "brain slipperiness", or analogising, to solve. You need to consider the information provided and see the connections that are not obvious at first glance. Different puzzles work differently, of course, but many of them involve a level of analogising that stretches the normal skill level of most people.

But the thing which brought this topic to my mind recently was not solving puzzles, but the puzzle creation process, which we have to go through before we can run the competition. One of the emergent traditions of our competition is that the fourth day of puzzles released (out of a total of five groups, released mostly at daily intervals) is something we call "mess with heads day". It started in 2010 with the set of puzzles we titled:

• Martial Law,
• Chebble,
• Blockhead, and
• The Missing Monkey.

Figure 1. The puzzle Martial Law, Chebble, Blockhead, and The Missing Monkey.

If you look at those four puzzles, you'll see the perhaps surprising detail that all of them present exactly the same image (reproduced on the right in Figure 1). The trick here is that there are in fact four more or less separate puzzles embedded in the same graphic, and that you can figure out what the answers are to each of them by associating the derived answers to the puzzle titles. (If you are curious and want to see what the solutions are, there are complete worked solutions linked from this page.)

In 2011 we decided to continue the "mess with heads" theme by making a set of "meta" level puzzles, which all made explicit references to the fact that they were puzzles. Like a TV show about a TV show, these were puzzles about puzzles. They were:

• A Typical Puzzle, which looks like a rant about how terrible modern puzzles are, and includes many of the elements it complains about (though designed in a way to be far less objectionable),
• Double Jeopardy, which looks like a worked puzzle solution that has been accidentally leaked,
• Team Statistics, which looks like a copy of the team rankings page as it stood roughly at the time the puzzle was released, and
• Puzzle Competition Reminder, which looks like the e-mail message we send to teams to remind them the competition is about to begin, with some other discussion about secret details of how one of the puzzles works inadvertently attached.

In 2012 we continued again with a set of four puzzles which resembled puzzles released earlier in the same competition, with the twist that they looked like similar puzzles but without the necessary clues needed to solve the puzzle. The puzzles were:

• Take More Sides, a crossword-like grid with coloured squares, which resembled the previous puzzle Take Sides, which was a crossword grid with coloured squares and a set of picture clues,
• Caverns, which presented a set of complex maze pieces that look like they need to be assembled jigsaw fashion, resembling the earlier puzzle Labyrinth, which presented a (much simpler looking) set of jigsaw-like maze pieces plus some additional information based on people in the maze that helped you figure out how to assemble the maze,
• Blaiseing Sun, a pyramid of numbers, which resembled the earlier Pyramid Trap, a similar pyramid of numbers plus a list of cryptic crossword style clues,
• Fabrication, a collection of strange pieces partly shaded sort of like crossword grids in odd orientations, which resembled the earlier Danger Looms, very much the same thing plus a set of actual crossword clues.

Figure 2. The puzzles Some Assembly Required, Interactions, Novel Plots, XYZ-Word, and Flags.

This year we continued the theme with our day 4 puzzles. This year we also expanded each set of puzzles from four to five puzzles. The five puzzles we published in this year's fourth group were:

• Some Assembly Required,
• Interactions,
• Novel Plots,
• XYZ-Word,
• Flags.

If you look at the puzzles (reproduced in small size in Figure 2, right) you'll notice that each one is a jumble of large geometric shapes which fit together to form a rectangle. Printed on the shapes are what look like parts of various different puzzles. It should be reasonably obvious that the first thing you need to do here is cut each puzzle into pieces along the lines and reassemble the pieces into the various different puzzles, jigsaw fashion. This is fairly easy to do, but the twist comes when you realise that the pieces you are given, even though distributed across five separate puzzles, reassemble to give only four different puzzles! Where is the fifth puzzle?!

For any of you who wish to try this for yourself, I won't reveal the answer here (but if you're curious and can't be bothered working it out yourself, there are worked solutions for all the puzzles here). That's not important to the point I want to make about analogies for today's discussion.

The point I want to make is related to the testing we do on our puzzles we've made before we release them in the competition. To ensure the puzzles are the best we can make them (including importantly that they can actually be solved by someone who didn't create them!), we test solve them without assistance from the creator, then provide feedback to help them tweak the puzzle to remove any annoying, frustrating, or effectively impossible elements.^[1] We also give feedback on creative elements, such as graphical design, layout, and the puzzle title. And this year we had a suggestion which struck me as very interesting.

The puzzle "Some Assembly Required" was originally titled "Tangrams", referencing the Chinese puzzle game. It seemed like a perfect name to me, to the puzzle creator who originally suggested it, and to most of our testing group. The allusion was meant to be that the geometrically shaped pieces in the puzzles had to be cut apart and reassembled into other shapes - the very essence of the original Chinese tangram puzzle.

Cut into geometric shapes. Though this is probably not tangrams.

Things were fine for a day or two, until the final member of our testing group saw the puzzle. He liked the puzzle... but he objected, quite strongly, the name "Tangrams". His objection was that the pieces were not tangrams. He said that "tangram" is the very specific dissection of a square into exactly seven pieces of predetermined and invariable shapes. A tangram was not a dissection of a rectangular shape into the differently shaped pieces we had used in our puzzle.

I was somewhat taken aback. Sure, the puzzle pieces were not exactly the same tangram shapes of the traditional Chinese puzzle game, but they were pieces that you cut apart and assembled into different shapes, which is what a tangram is all about. There were so many similarities between traditional Chinese tangrams and what we were doing in our puzzle that I felt it was perfectly justified to title the puzzle "Tangrams". Our last tester disagreed.

What I was doing in my mind was drawing an analogy. Tangrams and our puzzle both had a large geometric shape cut into several smaller geometric shapes, and each embodied a puzzle requiring reassembling those shapes into a different geometric shape. My mind saw all the links and easily made the analogy: "This is just like a tangram!" And so I was happy with the title. Our final tester was given exactly the same information I was given, but his mind rejected the analogy to tangrams. In his mind, tangrams have some very specific property that our puzzle did not, and so his mind refused to make the link that my mind made. And so he objected to the title "Tangrams".

In the resulting discussion over whether we should change the puzzle title or not, I pointed out how the puzzles shared so many features with tangrams that it would be almost silly not to call them tangrams. It was just a matter of applying the definition of the word a little more loosely than normal, which I was perfectly happy to do. Unfortunately by this time some of the others had considered the arguments and decided that if we could change the title to something that we were all happy with, then that would improve the situation. We floated some suggestions, and "Some Assembly Required" turned up. I was happy enough with that, so it was adopted, although I still think "Tangrams" was a superior title.

These plants get food by catching it. The analogy here is to someone who traps animals.

So anyway, the point here is that some people make some sorts of analogies more easily. I'm not saying our tester who objected to "Tangrams" is bad at making mental analogies. What really happened was that in this case he had a very deep knowledge of the subject matter (tangrams) and he had progressed beyond the point where his mental picture of tangrams was "cut something into pieces and make new shapes out of the pieces" to "tangrams are a specific dissection of a square into the seven shapes I am familiar with". It's the same as if a biology graduate was told that "photosynthesis is where a plant takes sunlight and water and air and makes food for itself". She would say, "No, that's not really right at all! It's actually a biochemical process in which...." Yet the former explanation is exactly what we use when teaching young children how plants grow, and it serves their purposes just fine.

So our minds make analogies all the time, and some of us are better at making different analogies than others. I suspect there is also another distinction between people who make analogies more or less easily. As I understand it, people with varying degrees of autism tend to be more literal-minded when dealing with the world and interpreting new information. I am by no means an expert on autism, and I may well be completely off the wall here, but this sounds to me like an inherent difficulty in making analogies. Many of us deal with new things by connecting them to things we know about: "this situation is like this other situation I'm comfortable with, so I know how to react". If the analogy-making process in the brain is less capable of making these sorts of connections, the result could easily end up leaving someone foundering in a new situation, because they don't see how it resembles a more familiar situation.

I'm now hypothesising wildly and may be completely wrong, but it strikes me as an interesting thought.

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[1] Although our testing process is fairly rigorous, it's not perfect. Unfortunately sometimes a puzzle slips through that we happened to test solve without too much trouble, but which other people stumble on because something we thought was easy turns out to be far less easy when subject to the glaring light of hundreds of solvers and massed statistics. Alas, our testing cannot extend to testing each puzzle on several hundred solvers before the competition itself.