The Rake Court is very large compared to the Tug Court and Reflection Pools, and consists of 135 tiles in a rectangle format, 9 tiles high by 15 tiles wide. The Rake will start offset one tile east or west from the dead center of the court, skewed in favor of whoever is going second. For purposes of this guide, it will be assumed that the Rake starts one tile east of dead center. With this setup, Player 1 is going West, Player 2 is going East. So, as soon as the game starts, the court looks like the diagram below, with no mushrooms on the court. The "Rake" is the white statue on the court.
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To win in Heket's Rake, you want to move the Rake to the farthest column on your side of the court, or past it. To determine which side of the court is yours, switch to Cartographer's Camera (I also find this camera view to be the best for playing Heket's Rake) and click on the desk. The side of the desk window that your name appears on matches up with your side of the court from Cartographer's Camera.
In Rake, you and your opponent alternate turns. On your turn you may either plant a mushroom, or move the rake. If you choose to plant a mushroom, it plants the mushroom on the tile where you are standing, and your turn ends immediately. To move the rake, you must "jump" it over one or more mushrooms (it doesn't matter who placed the mushroom), and then from that tile, you can "jump" it again. Jumping over a mushroom removes the mushroom from the court. If you're moving the rake, your turn ends when there are no mushrooms adjacent to the Rake, or you click the "Pass" button. (If there is a possible spot you could jump to, but you don't want to, you'll have to hit Pass to end your turn. Unlike Tug, there is no time limit per turn, and so games between two experienced (or horribly inexperienced) players can often last an hour or more. Note that you can move off the court's east or west side, but not off the court's north or south side.
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In the above diagram, Player 1 has spent his three turns placing mushrooms directly towards his side, while Player 2 has spent his two turns doing the same and it is Player 2's turn. By jumping the two shrooms east of the Rake, Player 2 has effectively won the game.
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No matter where Player 1 places his mushroom, Player 2 can respond by placing his mushroom directly east of the Rake.
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This forces Player 1 to jump his one shroom, wasting his turn.
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Now Player 2 places a shroom east of the Rake's new position, and there's nothing Player 1 can do to stop him.
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By placing a mushroom at E5, Player 1 stops Player 2 from linking up with his chain of mushrooms, at least in the short term:
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There's a lot of mushrooms between the Rake and Player 2's side, and Player 1 will have an increasingly difficult time stopping Player 2 from linking up. While Player One could jump up and left to E3, this isn't much help, because with only one shroom at F4, Player 2 has linked up with a mushroom chain towards his winning column. However, Player 1 could make the following, complicated jump:
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Now, a win by Player 1 is inevitable (barring any screw-ups like a misclick).
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If Player 2 doesn't want to extend his chain to the east out and force Player 1 to move, he could instead plant at E1, preventing Player 1 from jumping there. (not something I'd necessarily recommend, but just meant to show the mechanics of extending a diagonal line):
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Now, Player 1 plants at F3, trying to create an alternate chain. Player 2 responds by planting at G3:
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Now, Player 1 can't directly use the northwest mushroom chain at all, and must start anew. Since Player 2 already has one mushroom towards his side, he's at an advantage.