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The power in question is Final Rebuke, a Paladin level 9 power from Divine Powers:

Daily • Divine, Reliable, Weapon
Standard Action
Melee weapon
Target: One creature
Attack: Strength vs. Fortitude
Hit: 2 [W] + Strength modifier damage, and you push the target 5 squares. If the target ends this movement in a square adjacent to a solid obstacle, the target takes 1 [W] extra damage.

Legend: The B is the user of the attack, E the target, the O is a wall, and the Xs are empty squares.

When you use and hit with the attack:

XXXXXXOX
BEXXXXXO
XXXXXXXXO

… then the target ends his/her forced movement but does not hit the wall:

XXXXXXOX
BXXXXXEO
XXXXXXXXO

… will the target take the 1[W] extra damage or not?

And then will it work if the solid obstacle is to the left and/or right?

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Adjacent, aside from the standard english definition of "next to", has defined rules, as per page 201 of the Rules Compendium:

Adjacent Squares: Two squares are adjacent is a side or corner of one touches a side of corner of the other. Two creatures or objects are adjacent if one of them is in a square adjacent to a square occupied or filled by the other, or if they are in the same square.

In your example, the target would take the extra 1[W] damage because they are adjacent to the squares of a solid object.

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