You will need a regular mirror, some duct tape, and a reasonably opaque cloth, like a dark-colored T-shirt.
Tape the T-shirt over the light in your room, so that only a little bit of light can get out. Let your eyes adjust to the dark, and look in the mirror.
Your reflection is now in black and white.
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Or would you like to do it without reducing the amount of light in the room?
Your reflection appears in full color because light of every color is bouncing off of your body, off the mirror, and then into your eyes. If you want a black and white reflection, you need to interrupt the process at any point. Here are some possibilities:
1. Illuminate your body with monochromatic light.
2. Ensure that only monochromatic light is reflected from your body.
3. Ensure that only monochromatic light is reflected off of the mirror.
4. Ensure that only monochromatic light enters your eyes.
5. Ensure that only monochromatic light is converted to an image which is transmitted to your brain.
My flippant answer above used (5): I suggested a way in which you could shut down your color-perceiving cone cells so that only your black-and-white rod cells worked. We all see in black-and-white, in low-light conditions.
To perform a variant on (2), paint yourself white and wear a black shirt. Then your reflection would, in fact, be black and white.
The problem with all of these methods is that 'monochromatic' means one color, and grey is not one of your choices. Light only comes in the colors of the rainbow, red through violet. (And beyond, but we can't see those wavelengths.) Illuminating yourself with red light, or putting a good red filter on the mirror (or on your glasses) would give you a monochromatic, value-only image--the kind that you could use Photoshop to convert to what you think of as a black-and-white image. But seeing in red-and-black isn't as cool as seeing in white-and-black, is it?
Your 'mirror' could be a web camera, hooked up to a flat panel monitor. Not cheap, but doable. If it had a dedicated processor, it could run so fast you wouldn't be able to tell the difference.
If you don't want to use a computer or a camera, here's my final attempt:
Construct an array of photovoltaic cells, as small as possible. (These are semiconductor devices which produce a voltage proportional to the amount of light incident on their surface.) Isolate each diode by placing it at the end of a reasonably long (0.5 cm) tube pointing straight forward (all of these tubes could be integrated into one grid or honey-comb-shaped divider.)
Now connect each photodiode to a white LED, so that the amount of light produced by each LED is proportional to the voltage on its respective cell, and thus to the intensity of the light coming from (being reflected from) the user's face. The image would look an awful lot like a computer image, especially if the resolution were poor. But if you could make small enough cells, and pack enough of them onto one surface, then you'd have a reasonably convincing effect.
However, photodiode efficiency is about 10 percent, and so is LED efficiency, so only (0.1)*(0.1) = 0.01 = one percent of the light of the light energy that hit the 'mirror' would actually get reflected back at you. So it would be a black and white picture, but mostly black.
P