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From page 250 and 251 of the DMG:

To determine whether a target has cover against an attack or other effect on a grid, choose a corner of the attacker's space... trace imaginary lines from that corner to every corner of any one square the target occupies. If one or two of those lines are blocked by an obstacle (including another creature), the target has half cover...

grid diagrams taken from the DMG. The final diagram shows the construction lines provided in the DMG to determine cover (showing 3/4th cover) as well as the question askers own construction lines, superimposed showing it should be half cover.

It looks like they just chose the wrong corner. If the attacker was moved one square to the north it would be 3/4 cover. Or do you think they meant "choose the closest corner"?

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  • \$\begingroup\$ From my comment here: Similarly, in the half cover example, if the attacker chose the top-right square of the target no lines would be obstructed. Both examples would be better served by having the attacker one space up on the grid. \$\endgroup\$ – smbailey Jul 16 '20 at 20:21
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Their example just illustrates their point, it does not imply that it is impossible to avoid 3/4 from that position, only from that corner

I believe this is just supposed to be an illustrative example. If you choose the corner they chose on that image, then the target has 3/4 cover, but if you choose the corner that you chose, then the target only has half cover.

I don't think their example was supposed to imply that the character has no way to get around the 3/4 cover in that example, simply that by choosing the corner that they chose, they can show you what 3/4 cover looks like (on a grid).

That said, your observation is a good one; they might have been better off choosing the square above so that 3/4 cover is the best you can achieve from that position.

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  • \$\begingroup\$ What significance is there in picking a corner during battle? Is it mathematically possible to only have 3/4th cover? \$\endgroup\$ – AncientSwordRage Jul 16 '20 at 10:54
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    \$\begingroup\$ There might also be any number of other reasons for not always picking the optimal corner of origin to target a particular creature, such as placing an aoe attack so it doesn't hit a friendly target or so it gets to more enemies, etc. \$\endgroup\$ – Sdjz Jul 16 '20 at 10:56

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