Why does non-magnetic metal attract to only one pole of a magnet?

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How does a non-magnetic piece of metal have an attraction to only one of the poles when you put it near a magnet (it tries to flip to the opposite side)?
 
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It doesn't. That's what non-magnetic means. Do you have a specific example that you can show us?
 
by non magnetic, I don't mean that it's not permeable. I only used that word to indicate that the piece of metal is not a magnet in itself.

It is just a very flat piece of permeable metal about W1mm, H3", L2cm. It will attach itself to the magnet on both sides when placed on the magnet on it's flat side. But when I hold the bar loosely between my fingers and hold the magnet over it, it always flips to the same side every time.
 
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Oh. Sounds like the metal piece has become mildly magnetized. Try picking up some light piece of ferrous metal with it (like iron filings).