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The description of the light cantrip says:

You touch one object that is no larger than 10 feet in any dimension. Until the spell ends, the object sheds bright light in a 20-foot radius and dim light for an additional 20 feet. The light can be colored as you like. Completely covering the object with something opaque blocks the light. The spell ends if you cast it again or dismiss it as an action.

What happens when that object is broken into two or more pieces? Do the pieces carry part (or the whole) of the intensity of the 20/20 light? And if each PC gets one piece, wouldn't that effectively create many light sources at the cost of one cantrip (1 action)?

What if the object is destroyed (e.g. piece of paper that's burned)? Would that effectively terminate the spell?

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Destroying an object entirely would destroy the light, because the object no longer exists.

Strictly, the rules do not say what happens when you try to split a light, which means it's up to the DM to adjudicate this situtation.

There's no reason that says part of an object would emit less light. The spell emits the same radius of light whether it's cast on a small object or large one. But breaking the object to get two light sources seems to go against the intent of the rules, which is to create a single light source; you could otherwise illuminate an entire dungeon by lighting up a cake and dropping crumbs as you go.

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In combat, this distinction could make a difference because of action economy, but Light is a cantrip. So it seems like, in any other situation, it doesn't really matter if you break the thing before or after casting the cantrip. Allowing someone to split an item to get multiple lights from one casting doesn't seem all that big of a deal. A reasonable way to handle this if you don't want them to be able to multiply their lights is the make the light stay with the larger portion or to allow the player to pick which part keeps the light.

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