Visual Analogies, Series 1 Discussion

Now that you've tried the problems, let me present some of my answers along with a discussion of the how I got them. I'm going to borrow an idea from Fluid Analogies and present first a "concept web"; that is, a diagram of concepts raised in understanding and solving the puzzles.

[concept web legend]

To the right is the legend for the concept web diagram below. There are three kinds of objects: concepts, attributes, and links. Attributes are things that parts of each puzzle may be: inside, left, composite, etc. Concepts are more abstract. Position, for example, relates to attributes such as "outside" and "left". Links denote the relationships between attributes and/or concepts. They come in two types: related and opposite. For example, there's a related link between the attribute "inside" and the concept "position". Inside-ness is a type of position. The link between inside and outside is an opposite link, since the two are opposites.

When solving a problem, certain concepts, attributes and links are more active than others. If all the puzzles to be solved were black and white, then a concept like "color" wouldn't be active. Any set of analogy problems define a certain set of concepts, attributes and links to be relevant. Part of this set is below.

[concept web]

This is of course only a subset of all the attributes, concepts and links that this puzzle brings up. (Early versions of this diagram looked very different!) For example, one concept that isn't here is "containership". That might be linked to the attributes "contains" and "is-contained-by".

In fact, the first puzzle itself touches a concept that isn't shown: shape. When a circle is replaced with a square, the most salient concept is shape. Symmetry plays a role as well, but we're not conscious of it until we experience something that breaks the rules we've seen so far.

My answer to the first puzzle is to fill in the right half of the square. For the second puzzle, I see two solutions: filling in the "right half" of the triangle (everything to the right of the top vertex), or filling in each corner. Neither is a truly satisfying solution.

The third puzzle brings up another concept not in this web: addition/removal. My answer to this one is counterintuitive. The "obvious" answer is to remove the right half, leaving a mirror image of the example's answer. Yet when originally writing this paragraph, without envisioning what I was writing, I came up with the opposite answer! The answer is the same as the example's answer: a disc missing the left interior half. Here's the explanation I wrote:

[My answer is] to remove the left half of the disc's interior. My reasoning is that adding to an empty space is the analogical mirror-image of removing from a full space. Since the concept of "opposite" is operating so strongly, it spills over into the left/right attributes, and we reverse everything. Removing the right half of the disc's interior ignores all the reversals taking place, and feels to me like an incomplete solution.

I'm amazed that my intuitive answer and my reasoned answer are completely different - and I like both of them.

Puzzle 4 explores the concept of composition. When the single circle becomes two, a whole new set of pressures is introduced. Are the two circles a single unit ("single") or a group of two independent sub-units ("composite")? Each leads to a different solution: (a) fill in all of the right circle; (b) fill in the right half of both circles. I find both of these to be good solutions. However, that equanimity begins to break down in the next puzzle. As we lose the symmetry of two circles, the single/composite pressures change.

The rules really seem to go out the window in the sixth puzzle. Suddenly we are bereft of circularity, containership, and almost every concept except symmetry.

I don't have a good answer for this; it's more a reminder that any analogy can be pushed too far. That's certainly the case here. Two dimensional images are an incredibly flexible medium for the creation of puzzles; there are so many degrees of freedom that they become hard to analyze. Perhaps it would be better to add some restrictions. Not tonight. This started as a "What are your answers?" experiment with friends. It's been analyzed enough for now.

And with that, it's probably time to call an end to this, and to go re-read parts of Fluid Analogies again.

Last updated 4 June 2000
All contents ©1998-2002 Mark L. Irons

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