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Some good ideas there about how to remove a screw. My own way is to place the item under a drill press, with a screwdriver bit in the drill chuck. Press down on the chuck handle,then insert a shaft into the holes in the chuck and while pressing hard, undo the screw by turning the chuck by hand.You can also try heating the screw head with a soldering iron, which cause the screw to loosen its grip.Another idea, for impossible screws is to weld a rod to the screw head, then extract rod and screw together.
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Beyond Beginner Welding
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Another way is to print your mirrored pattern on transparent acetate sheet. Then apply it, face down, to the wood and squeegee it all over with a credit card, or similar.You do need to make sure it doesn't move, so I use sticky tape to hold it in place. On some surfaces, you may need to do it twice, making sure the two patterns line up, which isn't difficult, because the sheet is transparent.Since the inks doesn't get absorbed by the acetate sheet, you can use the same piece over and over again, by just rinsing it under the tap.