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I was wondering what exactly does X-foot cube mean in a spells range description.

Let's look Thunderwave for example. It says:

A wave of thunderous force sweeps out from you. Each creature in a 15-foot cube originating from you must make a Constitution saving throw.

We took that to mean, all points a square 15 feet from the caster. Or does it mean a 15x15 foot square with the caster in the middle?

Let P be the player and X a 5 foot block that the spell affects.

We decided it to be:                    Or is it Really:

     xxxxxxx                        xxx
     xxxxxxx                        xPx 
     xxxxxxx                        xxx  
     xxxPxxx
     xxxxxxx
     xxxxxxx
     xxxxxxx
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It's not quite your first option, but you do have a lot more choices than your second option. In particular, the caster does not need to be at the center of the cube.

15 foot cube means 15 feet is the length of a side. "Originating from" puts the origin anywhere on any face of the cube, which I take to mean it could be along the bottom side or on a corner (or mere inches from a corner).

So it could be any of these options, or slide it around pretty much anywhere, so long as the cube is 15x15x15. And if you're not working with a grid, go ahead and rotate it however you want.

XXX  XXX  XXX
XXX  XXX  XXX
XXX  XXX  XXX
   P  P  P

XXX  XXX  XXX
XXXP XPX PXXX
XXX  XXX  XXX

   P  P  P
XXX  XXX  XXX
XXX  XXX  XXX
XXX  XXX  XXX

Note that the center option would work if you put the point of origin along the bottom face of the cube.

For the exact text, see page 204 of the PHB.

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It's the second scenario. The answer is in your own question.

A wave of thunderous force sweeps out from you. Each creature in a 15-foot cube originating from you must make a Constitution saving throw.

... Each creature inside a 15 foot cube....

That's it. The cube is 15 feet.

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    \$\begingroup\$ pp.204 rules on cube shows the point of origin as the side of a cube, rather than the center, so wouldn't the spell be cast like P[_ _ _] rather than [ _P _ ] ? \$\endgroup\$ – Orvir Feb 18 '15 at 4:23
  • \$\begingroup\$ according to p204, yes I would say you are correct. \$\endgroup\$ – Escoce Feb 18 '15 at 18:29
  • \$\begingroup\$ It also states that the player can choose to make the source square in their own square. So I think technically, it could be either. \$\endgroup\$ – Constuntine Aug 10 '15 at 19:15

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