Problem 3
On an infinite chessboard, start with pieces in an by block, one per square. A move jumps horizontally or vertically over an adjacent occupied square to the unoccupied square immediately beyond, removing the jumped piece. Find those values of for which the game can end with only one piece remaining.
Step 3 of 4: Use the strip lemma
In plain words
A three-row gadget consumes a strip without changing the shape needed for the next gadget.
Detailed analysis
The constructive part on the AoPS page uses a local strip reduction: in a filled 3 by m strip, perform jumps from one boundary inward in the alternating horizontal and vertical pattern. The strip can be replaced by its boundary row while preserving the ability to continue the same reduction on the adjoining rectangle. Rotating the pattern handles both orientations.