According to the best of my knowledge, "with" is the answer since "with" is used to mention what was used to do something while "by" is used when to mention who did that thing.

eg. The door was opened "with" a key by Smith.

Here "with" is used with the tool used to open the door ie "key" while "by" is used with doer of the action.

I hope this helps.
"By a brick" would be correct here. With the passive voice, "by" indicates the agent doing the action, in this case the brick which was thrown by a hooligan. "By" goes with many verbs in the passive voice, including "break." In this case, the brick breaks the window by accident; the brick was not meant to be used as a tool.

"With a brick" would indicate that the brick is a tool. The tool would be something normally used on purpose for a particular task. We'd say that someone eats ice cream with a spoon, opens the door with a key, and cooled the drink quickly with ice:

"¢ Ice cream is usually eaten with a spoon.
"¢ The door can be opened with a special key.
"¢ The coffee can be cooled quickly with a lot of ice, but the coffee will be diluted.

You might say this:

"¢ The window was broken with a special tool used by a professional locksmith.
"¢ The window was broken with an ax that the firefighters used.

On the other hand, when the window is broken not by a special instrument -- by accident or by nature -- you would say "broken by":

"¢ The window was broken by a large hailstone.
"¢ The window was accidentally broken by a baseball that the kids were playing with.
"¢ The window was broken by the branch of the tree that fell down in the storm.
"¢ The window was broken by a brick thrown by a hooligan.

"With" is normally used in the passive voice when you use a special instrument with a purpose: with an ax, with a fork, with chopsticks, with my fingers. "By" is used for many things: run over by a truck, written by Hemingway, attended by thousands of people, broken by a badly aimed football.

The sentence in question -- the window was broken with /by a brick thrown by a hooligan -- seems to need "by." Probably the window was broken when the hooligan hurled it from a place not too near the window. The hooligan didn't think beforehand about using the brick as a tool.

If the hooligan had brought the brick with him and intended to use it as a tool, then we could use "with." The sentence would be something like this:

The window was broken with a brick that the hooligan had brought with him especially for the purpose of breaking windows.


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