Remember Your SATs Analogies are useful but imprecise. Without knowing the context in which an analogy is used, a listener can find that even the simplest statement can be ambiguous. If Shelley says, "I'm going to pay for my beer now," and Tim replies, "Me, too," what does he mean? Perhaps he, too, intends to pay for beer. Or maybe he's about to pay for a soda and is mentally equating beer and soda.

Douglas Hofstadter, author of *Gödel, Escher, Bach* (Basic Books, 1979), has been working with graduate students at Indiana University and the University of Michigan to develop computer programs that solve analogy problems. The goal is to model the twists and turns of human thought, not merely to compute a single right answer. The puzzles that follow are based on concepts found in Hofstadter's *Fluid Concepts and Creative Analogies* (Basic Books, 1995).

NUMEROUS ANALOGIES The computer program Seek-Whence was Hofstadter's first attempt to emulate human analogy-making. Developed with then grad student Marsha Meredith, Seek-Whence tries to continue a given sequence of numbers. Such problems usually have many plausible answers. For instance, what is the next number in the sequence 1 2 1 2 3 . . . ? If you group the sequence as (1 2) (1 2 3), then it might continue . . . (1 2 3 4) (1 2 3 4 5). But if it's grouped as (1 2) (1 2) (3 . . . ), then it could continue (. . . 4) (3 4) (5 6) (5 6).

The sequences below use simple mathematical operations, such as adding or subtracting 1, alternation, and grouping. Other operations, such as multiplication or exponentiation, are forbidden. Can you explain the logic behind the three continuations for each of the six sequences below?

**1.** 1 1 2 1 1 2 . . . **a)** . . . 3 1 1 2 3 4 **b)** . . . 3 2 1 1 2 3 **c)** . . . 1 1 2 1 1 2

**2.** 1 0 0 1 1 1 . . . **a)** . . . 0 0 0 0 1 1 1 1 **b)** . . . 1 1 1 2 2 1 1 3 **c)** . . . 1 2 2 1 3 3 1 4

**3.** 2 1 2 2 2 3 . . . **a)** . . . 2 4 2 5 2 6 **b)** . . . 2 3 4 2 4 5 **c)** . . . 4 4 2 5 6 6

**4.** 1 1 2 1 2 3 . . . **a)** . . . 1 3 4 1 4 5 1 **b)** . . . 1 2 3 4 1 2 3 **c)** . . . 2 2 3 4 3 3 3

**5.** 1 2 2 3 3 4 . . . **a)** . . . 4 5 5 6 6 7 7 8 8 **b)** . . . 4 4 5 5 5 6 6 6 6 **c)** . . . 5 6 6 7 7 8 9 10 10

**6.** 2 1 1 2 1 2 1 2 1 1 2 . . . **a)** . . . 1 2 1 2 1 2 1 2 1 1 2 1 **b)** . . . 1 2 1 1 2 1 2 1 1 2 1 2 **c)** . . . 2 2 1 1 2 1 2 1 2 1 1 2

BE GLAD THESE WEREN'T ON YOUR TEST

Hofstadter's most ambitious program is Letter Spirit, developed by then grad students Gary McGraw and John Rehling. Given several lowercase letters of a font, Letter Spirit generates other letters of the alphabet in the same style. To keep the problem manageable, the program draws all letters on a coarse grid of line segments as shown at right.

Above are the first five letters of a human-designed grid font called Standard Square. Below are the first four letters from seven other grid fonts. Can you draw the e that goes with each font? Consider both style and legibility of the letters as you attempt to figure out the visual theme the font designers had in mind.

LETTERS ARE TO LOGIC AS ABC IS TO . . . ? Copycat, a software program developed by Hofstadter's former grad students Melanie Mitchell and Jim Marshall, solves analogy problems of the form "ABC is to ABD as XYZ is to what?" Instead of isolating just one solution, the program is designed to find roughly the same solutions that real people do. Many people answer the "ABD" analogy above with "XYA," reasoning that A follows Z just as D follows C. Others suggest "XYY," changing Z into an adjacent letter. Another answer is XYD. WYZ is a similar, but subtler, solution.

Likewise, each of the analogy problems at right has more than one solution. Find at least two answers per problem. Typically, one answer will be more clever than the other. Copycat considers only the order of letters in the alphabet and ignores their shapes.

**1.** AABC is to AABD as IJKK is to _____ **2.** MNO is to MNP as MRRJJJ is to _____ **3.** EFG is to DFG as GHI is to _____ **4.** EQE is to QEQ as RVVVR is to _____ **5.** FG is to GH as FFG is to _____ **6.** GFF is to GGF as SSSTT is to _____

**Solution**

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For more information, see Douglas Hofstadter's *Fluid Concepts and Creative Analogies* (New York: Basic Books, 1996). Hofstadter's research group, the Center for Research on Concepts and Cognition, can be found at www.cogsci.indiana.edu.

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