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Difference between Come up against a brick wall and Run one’s head into a brick wall

come up against a brick wall —(also: come up against a blank wall) encounter an insurmountable obstacle or difficulty:

  • So far, our efforts to stop the smuggling of protected species have come up against a brick wall.

Note: The expression is not equivalent in meaning to the phrase have one’s back against the wall—(also: be up against the wall) have run out of options; be pushed to the last extremity:

  • I don’t give up easy and have had my back against the wall before. I don’t intend to give up without a battle.

run one’s head into a brick wall —(coll.) persist with smth. without making any progress (also: run one’s head into a stone wall):

  • Trying to persuade him to change his plans is like running your head into a brick wall.