Eruvin 76A

זה נעשה בית שער לזה, וזה נעשה בית שער לזה. אמצעי הוה ליה בית שמניחין בו עירוב, ואין צריך ליתן את הפת.

this outer house becomes a gatehouse to this courtyard, and that outer house becomes a gatehouse to that courtyard, and therefore the residents of the outer houses need not contribute to the eiruv. The middle house between them is the house in which the eiruv is placed, and therefore its residents need not contribute bread for the eiruv.

בדיק להו רחבה לרבנן: שתי חצרות ושני בתים ביניהם, זה בא דרך זה ונתן עירובו בזה, וזה בא דרך זה ונתן עירובו בזה, קנו עירוב או לא? מי משוית להו לגבי דהאי בית, ולגבי דהאי בית שער [ולגבי דהאי בית שער, ולגבי דהאי בית]?

The Sage Raḥava tested the other rabbis: If there were two courtyards and two houses between them, and a resident of this courtyard came through this house that opens to his courtyard and placed his eiruv in that house farther from his courtyard, and a resident of this other courtyard came through this house that opens to his courtyard and placed his eiruv in that house that opens to the other courtyard, did they acquire the eiruv or not, i.e., are the two eiruvin valid? Do you render it a house with regard to this courtyard, whose eiruv was placed there, and a gatehouse with regard to that one who passed through it in order to place his eiruv in the other house? And similarly, do you render the other house a gatehouse with regard to this one and a house with regard to that one?

אמרו ליה: שניהן לא קנו עירוב, מה נפשך: אי בית שער משוית ליה, הנותן את עירובו בבית שער אכסדרה ומרפסת ⁠— אינו עירוב. אי בית משוית ליה ⁠— קא מטלטל לבית דלא מערב ליה.

The rabbis said to Raḥava: Neither of them has acquired his eiruv. Whichever way you look at it, it is difficult: If you consider either house a gatehouse, the halakha with regard to one who places his eiruv in a gatehouse, a porch, or a balcony, is that it is not a valid eiruv. And if you consider either one a house, he would be carrying into a house for which he is not establishing an eiruv. Since the assumption that benefits one of them harms the other, and there is no way to establish firmly the status of these houses, the residents of both courtyards fail to acquire their eiruv.