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Boil eggs by placing in large pot and filling with just enough water to cover. Allow eggs to boil for 10 minutes, then turn off the burner and allow to sit another 10 minutes.
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Meanwhile, prepare bowl of ice water. Once 10 minutes have passed, add the eggs to the ice water and allow to sit in the water 10-15 minutes, or until they're cool.
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Peel eggs under cool running water.
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Slice eggs in half lengthwise and place yolks in small bowl.
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Using a fork, mash egg yolks, mayonnaise, mustard, salt, and pepper in bowl. Set aside.
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Depending on the number of colors you'll be using, add 1 cup of boiling water and food coloring to each bowl.
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Add as many egg halves to bowl as will allow to be completely flat and completely submerged. I erred on the side of lighter more pastel colored eggs, but if you want deeper colors you can allow to steep by lightly covering bowls with a paper towel.
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Remove eggs from bowl and allow excess coloring to dry on a paper towel.
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Spoon egg yolk filling into eggs and lightly sprinkle with paprika.
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Enjoy right away or chill for 30 minutes before serving.