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?

  • $\begingroup$ I think the first game is called "Conway's Soldiers". In this game, one peg can jump over another peg like you described, and there is also a famous proof that a peg can reach only the first four horizontal lines. $\endgroup$ – Toby Mak Oct 30 '17 at 23:41
  • $\begingroup$ Yep, that's the first one. Thanks! Want to shove that in an answer? $\endgroup$ – Brondahl Oct 31 '17 at 9:16

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