The ‘active ingredient’ of magic erasers is [Melamine](https://en.wikipedia.org/wiki/Melamine_foam). It is a surprisingly hard material that only feels soft, and when it’s a foam it has many abrasion surfaces to work with. So essentially it is a sandpaper that can get into cracks of anything that is harder than it (and it scratches off layers of anything softer than it). Once it scratches off the thing it’s cleaning, it crumbles away from the sponge while surrounding it; so greasy residues are caught and contained, kind of like a pencil eraser and graphite.
Edit since this got some attention, but I think people are focusing on the wrong part: Most aggressive cleaners are abrasive in some way (they are also known as [scouring powders](https://en.wikipedia.org/wiki/Scouring_powder)), so melamine isn’t special for that; instead, it’s because it’s a “reticulated foam” (that means “net like”; the ‘bubbles’ of the foam are only made out of the lines where they intersect, while there is still lots of open space). Because of this, both the hard melamine can get into the cracks as I mentioned above (the foam is flexible, even though it’s made out of hard stuff) and the mess can actually get into the foam instead of just being pushed away. Once the mess is inside the foam, the chunk of foam with it can break off and keep it from going back and being a mess. This second part is why melamine works in places that chemical cleaners or scouring powders have a tough time.
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