Heuristic vs. Analogy

There's a classic book on heuristics called How to Solve It   by George Pólya. Back when I was teaching, I often recommended it to my students. You are correct that heuristics is the "science" of general problem solving, and it can be applied to anything from mathematics to psychology to management.

Within the discipline of heuristics, several specific problem solving methods are identified. Using analogy is just one specific method. Others are simplification, restatement, visualization, reductio ad absurdum, trial and error, etc. If I recall correctly, there's about a dozen specific methods mentioned in his book.

So the answer to your question is: Analogy is a subset (or component) of heuristics.

The Mutilated Chessboard Problem is a great example of heuristics in action:

• a chessboard has 64 squares
• you have 31 dominoes, each of which covers exactly 2 squares of the chessboard
• you remove 2 diagonally opposite corner squares from the chessboard (leaving 62 squares)
• can the chessboard now be totally covered by those 31 dominoes, without cutting any dominoes?

Using the trial and error method, you could spend a huge amount of time looking for a solution (and wouldn't find one). Removing two diagonally opposite corner squares from a chessboard removes two squares of the same color. Each rectangular domino must cover two squares of opposite colors. Thus, it is impossible to cover the 62-square chessboard with the 31 dominoes. Pólya would cite this as an example of the visualization and restatement methods of problem solving.

Am sure you're right.

Still, why is that heuristic ?

Put another way , if I were to say it's not heuristic, where did I go wrong ?

You ask why "that" is heuristics. I assume by "that" you're referring to the method of solution for the Mutilated Chessboard Problem?

First, note that the MCP is only asking for an answer in the form of either "Yes" or "No."

Finding that answer is an example of heuristics because the method most people would try first (trial and error tiling of the board with dominoes) will fail to provide a solution. Success would immediately provide an answer of "Yes," but that ain't gonna happen. Failure would NOT provide an answer unless every possible tiling were tried (and that's a LOT of possible tiling arrangements).

After several such failures, it might start to seem this is a problem requiring some kind of deep mathematical insight related to geometry or combinatorics. But it doesn't, and that's where heuristics comes in.

The more observant solver would notice all failed attempts leave two squares of the same color when the 31st domino is in-hand (which is a hint to the solution). That's visualization, or perhaps pattern recognition.

So then we restate the problem as: Can I cover 32 white squares and 30 black squares with 31 dominoes? This would lead via basic arithmetic to the solution "No." At least it would if combined with the insight that each domino must cover both a white and a black square (more pattern recognition).

Pólya notes that we all use heuristics without realizing it. It's just an intellectual process engaged during problem solving. What he did in his book was formalize that process by giving names to the specific methods used, and providing examples of each. Granted, the names are somewhat arbitrary, and there's a lot of overlap between methods. But heuristics is an intellectual process, so things do get a bit fuzzy.

I'd like to suggest a different angle on the  MCP which, I think, better  illustrates a  heuristic.    I  suspect a person  would  not spend a huge amount of time using trial and error.   Instead,  after one or two trials,  one would use a heuristic: having experienced a couple of trials, it would  be evident that there is no way to cover the two squares that have no mate of the opposite color or to cover a row having 7 squares when each domino must cover 2 squares.   I think the key concept about heuristics is that such  decisions  are not backed up by  rigorous analysis  and  are not necessarily sound logically  or mathematically.

I'd like to address the original question, too, regarding how heuristics and analogy are different.   Both horses and dogs are herd animals.   If you had experience with horses, you know that where there are two or more horses, one must be the leader and the rest followers.   Without any experience with dogs (no heuristic) you could understand (using analogy) the behavior they engage in to establish the leader.   I suppose one could argue that the experience with horses provides a heuristic to understanding dog behavior but I'd argue that  such a  heuristic is only applicable due to the analogous herd behavior.