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Wall of Roots puts a "-0/-1" counter on itself to use its ability. How does this interact with a +1/+1 counter? Do they cancel out to make a +1/+0 counter? Does they cancel out the same as a +1/+1 and -1/-1 counter and both counters cease to exist? Or do they both ignore each other completely?

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They ignore each other completely, so the Wall of Roots will have both a -0/-1 counter and a +1/+1 counter (which has the net effect of having +1/+0).

The only reason +1/+1 and -1/-1 counters interact with each other is because of this state-based action:

704.5q If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it, where N is the smaller of the number of +1/+1 and -1/-1 counters on it.

There is no similar rule for -0/-1 counters.

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The only two power/toughness modifier counters which explicitly annihilate each other are +1/+1 and -1/-1 counters:

122.3. If a permanent has both a +1/+1 counter and a -1/-1 counter on it, N +1/+1 and N -1/-1 counters are removed from it as a state-based action, where N is the smaller of the number of +1/+1 and -1/-1 counters on it. See rule 704.

Other counters ignore each other and contribute to the power/toughness of the creature as they would do without each other. A Wall of Roots with one -0/-1 counter and one +1/+1 has power/toughness 1/5.

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