Materials
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I started with an edge grain cutting board that I had already made from alternating strips of cherry and maple.
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I also created some strips of wood that I would use for the inlay. The wood has to be thin enough so that you can bend it along the curve without breaking it. Therefore, I used multiple thin strips that were roughly 1/8" thick or less. You can use whatever material you like. I used a combination of walnut, maple, and padauk.
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You will need some MDF to create templates for the curves that you will cut. I made three curves, so that required at least three pieces of MDF: one for each curve.
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Waterproof glue that is food safe. I used Titebond III
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Cutting board oil (mineral oil). I used this one: Howard Products Butcher Block and Cutting Board Oil

Tools
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Router with:
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one or more dado bits. These are the ones that I used: Whiteside Undersize Plywood Dado 3 Piece Set
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flush trim bit. This is the one that I used: Freud 1/2" Flush Trim Bit with Bearing
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round over bit. This is the one that I used: Freud 1/4" Round Over Bit
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Band saw (or a jig saw)
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A few clamps to clamp up the cutting board
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Random Orbit Sander
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Sand paper (120-grit, 150-grit, and 220-grit)
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A drum sander is definitely helpful if you have one, but not absolutely necessary