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If I change computers and use the 12 word seed to use my wallet, how will the client know which private key I used in the previous computer?

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A long sequence of private keys can be computed from the seed in a standard way. The new client will import those keys for your use, while computing the corresponding addresses and checking the block chain to see which ones have been used. When it finds many keys in a row that haven't been used (perhaps several hundred; this number may be configurable in your client), it assumes it has found all the keys in the sequence that you actually used, and stops.

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