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Pipes draws the actual pipe dream on an n × n board, with 1, …, n across the top and w in one-line notation down the left. Each pipe joins equal numbers on these two sides. Empty cells have elbows and filled cells have a crossing shape. Two pipes cross at most once; where they meet again, the shape stays the same and the colors turn at the center.
Goal: move and create cells until the counter reaches its maximum.
Choose a cell with an empty square immediately to its right. Look upward through these two columns, skipping rows where both squares are filled. If the first remaining row has both squares empty, move the cell into its right square. A row with only one filled square blocks the move. After moving a cell, you can choose to create a cell in its original position. The green + stays available until adding a cell there is no longer legal. Cells that can be moved have a small circle at the top right.
Based on Chen-An Chou and Tianyi Yu, Constructing Maximal Pipedreams of Double Grothendieck Polynomials, Electron. J. Combin. 31 (2024), P3.15.