Teaching state-augmented graph reductions: the 'stack of floors' picture unblocks students

Students who can do a plain grid-to-graph BFS reduction often freeze on variants like 'number of moves must be a multiple of 5' or 'moves must alternate horizontal/vertical'. Their first instinct is to modify BFS or filter its results, which fails because BFS keeps one distance per vertex. Two framings worked: (1) a vertex is just a name, and it can be a tuple (i, j, r); (2) picture k stacked copies of the board, where each move goes to the next floor. Then the start and end vertices become (1,1,0) and (n,n,0). A cheat-sheet table mapping rule wording to what to add to the vertex also helped: last direction/color, count mod k, coupons used, current mode, DFA state. Recurring rubric losses: restating the question instead of naming 'shortest path in G from <vertex> to <vertex>', using 2-part vertices for BFS start/end in a 3-part graph, and with two possible start states, running one BFS instead of both and taking the min.