With regard to Tamiyo, Field Researcher's first loyalty ability, if one of the creatures I've selected deals damage both as an attacker (on my turn) and as a blocker (on my opponent's turn), do I draw a card each time? And if one of the creatures I've selected has double strike, do I draw two cards each time it attacks or blocks?
Tamiyo's first ability sets up a delayed triggered ability that triggers each time "either of those creatures [the ability refers to] deals combat damage" while that ability is active, not just once, because the ability has a duration, namely "[u]ntil your next turn" (C.R. 603.7b). Note that the delayed triggered ability applies only to certain combat damage, not any other kind of damage.
For the purposes of the delayed triggered ability, a creature with double strike generally deals combat damage at up to two different times during the combat phase (more specifically, in two separate combat damage steps) (C.R. 702.4b, 510.2), so that that ability will trigger each time that creature deals combat damage this way (and is either of the creatures referred to by Tamiyo's first ability) while that ability is active.
Yes. Every time the selected creatures deal combat damage you draw a card.
Note, that if double strike is involved, and your creature is blocked/blocking, the first strike may kill all adversaries and if so, the second strike doesn't happen, and neither does the second card draw.
For the purposes of the delayed triggered ability, a creature with double strike generally deals combat damage at up to two different times during the combat phase (more specifically, in two separate combat damage steps) (C.R. 702.4b, 510.2), so that that ability will trigger each time that creature deals combat damage this way (and is either of the creatures referred to by Tamiyo's first ability) while that ability is active.
Note, that if double strike is involved, and your creature is blocked/blocking, the first strike may kill all adversaries and if so, the second strike doesn't happen, and neither does the second card draw.
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