by Invariance is one of the advanced techniques. It requires the establishment of hypotheses. We choose empty cells (or cells containing Booleans) and successively set them to true and false, then observe the result.
Let A and B be two empty boxes or boxes containing booleans.
If, whatever the true or false state of A, B changes state and is always true or always false,
then B is invariant and can be replaced by its constant value true or false.
Since the February 2025 update, Boolean invariance is also calculated in implication. However, if implication is disabled (this is possible in the general editor), then Boolean invariance is still processed by the invariance routine. Note that invariance is always processed following implication, and is only performed if implication yields no results.
How is invariance performed on Actilud?
To keep the puzzles from being too difficult, the solver limits his deductions.
When the solver applies the invariance technique, it changes the value of an empty cell or one containing a boolean.
Only puzzles marked as often difficult are likely to use implication or invariance. In basic puzzles (jogging, fashion show, land yachting, orchard) the solver adds the square configuration; therefore, there is not systematically implication or invariance. Moreover, the difficulty may not appear because it depends on the order in which the propositions are treated.
In general, the search for invariance only takes place if the search for implication yields no results; cases of invariance are therefore quite rare. The solver starts by classifying the empty cells; it chooses in priority those which are in the neighborhood of booleans and those which are on rows containing false signs. For performance reasons, empty cells whose neighborhood is too sparse are ignored by the solver, which leaves – intentionally – a small probability of not being able to solve some puzzles.
Since the February 2025 update, the solver always favors solutions that require the least depth of analysis. They are thus -sometimes- more understandable and make the puzzles more motivating. In addition, the solver takes inconsistencies into account. To help you a little, the difficulty level of the implication and the invariance is indicated by the Advisor, when you use the solver in “step by step” mode. The level corresponds to the number of successful operations that had to be done to obtain a result. Level 1 is therefore the easiest: the invariance only required a single operation. Level 7 is the most difficult.
The operations performed are: intersections, complement, repercussion, fusion; the complement being executed multiple times. Intersection also includes consistency.