Basic puzzle aspect of counting, or progression of a well known sequence
Counting is both a simple puzzle aspect, as well as an ability. This is one of the foundation puzzle mechanics, either used directly in a puzzle piece, or as the basis for more complex aspects.
Counting refers to both summing up the number of items in a scene, or to a progression along known sequences.
Many puzzles require the player to count how many items are in a scene, or how many components there are of a puzzle piece.
For example: - the number of apples on a tree - the number of dots on a die - the number of people who went into a house
These types of puzzles are generally not referred to as math puzzles, even if they technically they are.
Counting could start from some offset, and may also involve subraction. For example, we may know 10 people started in the house, watched some leave, and need to know the count now. Again, these are generally not referred to as algebra puzzles, even if an equation could be constructed.
We're generally looking at the skill required to solve a puzzle, and if counting is all one needs, then it is considered a counting puzzle.
A core staple of puzzles is translation between a number and letter or symbol, based on a given, or known sequence. We call this sequence indexing, as we are indirectly refering to an item in a sequence by its index: the offset form the start of the sequence.
For example, take the numbers
16 21 26 26 12 5 map them to the respective count in the English alphabet. The 16th letter is P, so the first letter is P. Continuing, we get the word "puzzle".
This simplistic translation could be referred to as a "substitution cipher". But we generally don't think of this trivial form as a cipher. The player is not required to do any type of decoding work, or substitution analysis, typical of a cipher. Instead, they simply count along a known sequence, like the alphabet.
Sequence indexing are used for more than translation to words. The numbers may refer to the position of an item on a shelf, or to the mysterious symbols inscribed around a circle of an ornate door. Anything that can form a sequence can be indexed.
Sequences should have their own Grimoire entry, with indexing as a subsection of that. While counting is required, it is a different type of "thing".
Assume everything in this reference is a working draft, there's prone to be some mistakes and inconsistencies. I figure it's best to publish and get feedback rather than write for years in secret. The terms will change, the structure will shift, and the bugs will be chased out. It'll take a while.