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More specifically, how many tiles away do I have to be before I get bonus accuracy on my shotgun? Closest things I've found are this reddit thread and At what range do various weapons get range bonuses?. The latter is for XCOM 2. The reddit thread says,

Shotguns have about 50 aim bonus at 1 range, 0 bonus at 10 tiles, 29 penalty at 15 tiles.

But the UFOpaedia link is dead.

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9 tiles, put simply.

9 tiles straight is the range at which most weapons' aim changes. 6 diagonal, if I'm reading this Google Sheet correctly.

  • Shotguns get a proportional bonus if less than that distance. Proportional penalty if greater than that distance. Ranging from +52% to -40%.
  • Sniper rifles get a proportional penalty if less than that distance (up to -24%).
  • Rifles, LMGS, and pistols get a proportional bonus if less than that distance (up to +37%).

https://www.ufopaedia.org/index.php/Chance_to_Hit_(EU2012)#Weapon_Range

The Arc Thrower has a maximum range of 3 tiles. Its odds of success are 49/70/80/90% at 4/3/2/1 enemy HP respectively.

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