I'm sure I've come across 2 well-known recreational maths puzzles, both involving cells jumping over each other. Can anyone provide the relevant search terms?
The first has an fully-infinite grid, divided in two by a horizontal line. Every point below the line has a cell on it, and a cell can move by jumping 1 step over another cell, at which point the jumped cell is removed.
- Cells can only jump orthoganally
- Cells may only jump over a single other cell, which is immediately adjacent to it.
- Cells may only jump into an empty point.
The goal is to get a single cell as far above the initial line as possible, and IIRC the limit is 5?
The second starts with 3 dots in the 3 bottom-left-most points of a semi-infinite 2D grid i.e. $(\mathbb{N}^+\times\mathbb{N}^+)$
A dot can be replaced with 2 dots - one above and one to the left, as long as both grid points are currently empty. You can't ever get them fully out of those 3 initial cells. Proof involves defining an invariant where the value of a dot is 1/2^n where n is the New-York-Taxi metric from the origin.
What are the names of these ... puzzles?