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invariance principle

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Invariance Principle

(Note: This is a relatively imprecise concept so there is not a universally accepted terminology for it, but for the problems dealt with on this site we need a name for it.)

The invariance principle is a technique useful for figuring out the answer to a puzzle under the assumption that such an answer exists and is unique. It applies when a puzzle is not be completely determined by the constraints (which might be in the form of a diagram) and yet there is a unique answer. This means that there are multiple possible diagrams that can be drawn.

This invariance can be exploited in two ways, either to get an actual solution or as a route to solving the general case.

This is sometimes referred to as the Agg Invariance Principle due to the frequency with which it can be used to solve Catriona Agg’s geometry problems.

Strong Invariance

The strong invariance principle applies when there is a way to draw the diagram which makes it straightforward (or, at least, more straightforward) to see the solution. Often this involves exploiting a particular symmetry or coincidence of this special case.

Weak Invariance

If there isn’t a way to draw the diagram to make solution obvious, it still might be possible to use the variety of possible diagrams to reveal a route to a solution. This is the weak invariance principle. A common application is to mark on the diagram the possible locations of certain of the points in the diagram, these might be on a line or a circle. A good strategy for finding a solution to the general problem is then to prove directly that the shape is what it appears to be.

Revised on June 23, 2021 20:00:51 by Andrew Stacey
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