Given some transformation that acts on some objects, an invariant of the transformation is some measurement of the objects that doesn’t change once the transformation is applied. E.g. for the transformation that reflects a polygon in the plane over a given line, the area of that polygon is invariant under that transformation. A semi-invariant is similar, except instead of describing a measurement that doesn’t change, it describes a measurement that changes monotonically — either increasing or decreasing — with each application of the transformation. Noticing an invariant or semi-invariant can be tough.
Examples
A chocolate bar is a \(3 \times 4\) grid of small rectangles.
What is the minimum number of times you’d need
to snap a chocolate piece along a grid line
before you have \(12\) individual rectangles?
Don’t get distracted by the grid or all the possible ways to snap the chocolate bar. What happens each time you snap it?
The numbers \(1,2,3,\dotsc,30\) are written on a blackboard.
At each step we perform the following action:
choose any two of the numbers \(a\) and \(b\), erase them,
and then write the number \(|a-b|\) on the board.
Continuing this until only a single number remains,
is it possible that the last number is \(8?\)
What property of the sum of the numbers isn’t changing?
Start with the triple \((3, 4, 12)\).
You may take the following action to update the triple:
choose any two numbers within the triple, \(a\) and \(b,\)
and replace them by the numbers
\(\frac{4a}{5} + \frac{3b}{5}\)
and \(-\frac{3a}{5} + \frac{4b}{5}.\)
Can you reach the triple \((4, 6, 12)\)
by applying this action finitely many times?