Is it permissible for a thrice-divorced woman to live with her ex-husband in the same house for the sake of childcare, while avoiding seeing him?

If a husband divorces his wife with a final third divorce, or two divorces, or one divorce, and her waiting period (iddah) has ended, she becomes a stranger to him. It is not permissible for him to be in seclusion with her, touch her, or look at her. There is no difference whether the divorce was documented or not.

Shaykh al-Islam Ibn Taymiyyah said: "A woman divorced three times is a stranger to the man, like any other strange woman. So, it is not permissible for the man to be in seclusion with her, just as it is not permissible for him to be in seclusion with a strange woman. Nor is it permissible for him to look at her in ways he would not look at a strange woman. And he has no authority over her whatsoever."

Therefore, there is no harm if the divorced woman lives in a separate part of her ex-husband's house, provided that temptation is averted and he does not see her or enter upon her.