Definition
Invariant
An invariant is a quantity or property that stays unchanged during every allowed move or transformation.
Continue from this idea into connected AXIO math.
Defining relationship
The moves happen; one chosen fact refuses to move.
An invariant is a quantity or property that is preserved by each allowed operation in a specified process, transformation, or system.
- start5 counters
- + 27 counters
- + 29 counters
- − 27 counters
Every allowed move changes the count by 2, so odd stays odd.
12 is unreachable
What it means
First name the moves that are allowed. Then look for something those moves can never change, such as whether a number is odd or even. Tracking that fixed feature can rule out impossible results or shorten a search.
Remember it
The moves happen; one chosen fact refuses to move.
Example
Start with 5 counters. Each move may add 2 counters or remove 2 counters. The number of counters always stays odd, so reaching 12 counters is impossible. Odd parity is an invariant of the allowed moves.
Watch for
A feature is not an invariant just because it stays the same in one example; it must stay unchanged after every allowed operation.
Common mix-up
Always state both parts: the allowed operation and the property it preserves.
Exact meaning
An invariant is a quantity or property that is preserved by each allowed operation in a specified process, transformation, or system.