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The "illusion" is the idea behind Mirror Therapy for amputees -- the mirror makes it look like there are two limbs and the brain receives information that both limbs are intact and functional. A trial published in The New England Journal of Medicine found that it was effective for reducing phantom pain after 4 weeks of regular practice. I've heard people say it helps and others say it didn't. I've never tried it, since -- luckily -- phantom pain has not been an issue for me.