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I was playing MTG with my friend. I attacked with a 6/4 and he blocked with a 5/5. He says my 6/4 should die because its toughness is less than its block even though my creatures attack (a 6) is bigger than his creatures toughness (5).

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    Hint: what happens when a 5/5 blocks a 5/5? – Joshua Feb 1 at 17:05
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Unless one of those creatures has First Strike or Double Strike, they will deal combat damage simultaneously. The following will happen:

  • Your 6/4 will deal 6 damage to your opponent's 5/5. (Note that 6 ≥ 5.)
  • At the same time, your opponent's 5/5 will deal 5 damage to your 6/4. (Note that 5 ≥ 4.)
  • State based actions kick in, see that both creatures have damage greater than or equal to their toughness, and both creatures die.
  • "Protection from X" and "Trample" may also change the outcome of the fight. – Bernat Jan 30 at 15:52
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    @Bernat there are a number of other effects that could change the outcome, but Trample isn't one of them. – Caleth Jan 30 at 16:45
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    @Bernat Trample would change the effect of the fight, but in this context, "outcome" refers to what happens to the creatures. – Acccumulation Jan 30 at 19:06
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    @Bernat Magic is large enough that exhaustive lists of things that might be relevant are long and unwieldy. The relevant point in this answer is "deal combat damage simultaneously". Going down that route ends up with "... or unless one of the creatures is killed / bounced / frogged by an instant during the declare blockers step, ... " – Caleth Feb 1 at 17:13
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    @Caleth Every such effect would need to be handled before the Combat Damage Step. Only few effects happen during it. Namely: prevention, double strike, First strike and actions based on the state of a card, namely if put XXX, indestructible and regenerate – Trish Feb 4 at 23:17
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This stuff is explained very well in every core set starter box by about turn 3, usually having a 2/1 block a 1/1 or similar, but let's be frank and dig out the rules. Open at 510, we skip parts that are not relevant:

  1. Combat Damage Step

510.1a Each attacking creature and each blocking creature assigns combat damage equal to its power. Creatures that would assign 0 or less damage this way don’t assign combat damage at all.

510.1c A blocked creature assigns its combat damage to the creatures blocking it.[...]

510.1d A blocking creature assigns combat damage to the creatures it’s blocking.[...]

510.2. Second, all combat damage that’s been assigned is dealt simultaneously. This turn-based action doesn’t use the stack. No player has the chance to cast spells or activate abilities between the time combat damage is assigned and the time it’s dealt.

[...]

510.4. If at least one attacking or blocking creature has first strike (see rule 702.7) or double strike (see rule 702.4) as the combat damage step begins, the only creatures that assign combat damage in that step are those with first strike or double strike. After that step, instead of proceeding to the end of combat step, the phase gets a second combat damage step. The only creatures that assign combat damage in that step are the remaining attackers and blockers that had neither first strike nor double strike as the first combat damage step began, as well as the remaining attackers and blockers that currently have double strike. After that step, the phase proceeds to the end of combat step

Ok, what does that mean for us?

  • Step 510.1A & C say, that the 6/4 deals 6 damage to the 5/5 in absence of first strike, double strike or protection from the color/type of the 6/4 or an effect that prevents combat damage.
  • Step 510.1A & D say, that the 5/5 deals 5 damage to the 6/4 in absence of first strike, double strike or protection from the color/type of the 5/5 or an effect that prevents combat damage.
  • Step 510.2 says, we got a 6/4 with 5 damage and a 5/5 with 6 damage. In both cases, the damage is equal to or higher than their toughness.
    • Both creatures get destroyed1 2.
    • The destroyed creatures leave the battlefield3 and enter the graveyard4

1 - in absence of 702.12. Indestructible A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage
2 - 701.14. Regenerate can apply: “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.
3 - might trigger as a/this card leaves the battlefield effects
4 - might trigger as a/this card is put into the graveyard effects

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I think the source of your confusion is in what the stats of a creature are (I had the same confusion myself way back when I first started playing Magic).

A creature's stats are power/toughness, not attack/defense. So a 6/4 has 6 power and 4 toughness.

Power is how much damage it deals. Toughness is how much damage it takes to kill it. When asking which creatures die in combat, it does not matter which is the attacker and which is the defender. Each creature simultaneously deals damage to the other (based on its power), and then each creature that has taken damage greater than or equal to its toughness dies.

In the situation you give in the question, each creature has power greater than or equal to the toughness of the other, so they will kill each other (baring abilities like first strike, double strike, protection, regeneration, indestructible, etc).

  • Indestructible is a special case: it does take the damage but is not destroyed. see 702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g). – Trish Jan 31 at 14:23
  • @Trish the abilities I listed were merely a few examples of things that would make the statement "they kill each other" not true. Indestructible absolutely fits this criteria. – Zags Jan 31 at 14:25
  • To complete the listing in the end: 615.1. Some continuous effects are prevention effects. Like replacement effects (see rule 614), prevention effects apply continuously as events happen—they aren’t locked in ahead of time. Such effects watch for a damage event that would happen and completely or partially prevent the damage that would be dealt. They act like “shields” around whatever they’re affecting. – Trish Jan 31 at 14:25

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