What does single state of purity means in this hadith?

Question

Can you explain this highlighted line. This mean 3 people intercourse at the same time?

Sunan abu Dawud (2262)

Narrated Zayd ibn Arqam: I was sitting with the Prophet (peace_be_upon_him). A man came from the Yemen, and said: Three men from the people of the Yemen came to Ali, quarrelling about a child, asking him to give a decision. They had sexual intercourse with a woman during a single state of purity. He said to two of them: Give this child to this man (the third person) with pleasure. But they (refused and) cried loudly. Again he said to two of them: Give the child to the man (the third person) willingly. But they (refused and) cried loudly. He then said: You are quarrelsome partners. I shall cast lots among you; he who receives the lot, will acquire the child, and he shall pay two-thirds of the blood-money to both his companions. He then cast lots among them, and gave the child to the one who received the lot. The Apostle of Allah (peace_be_upon_him) laughed so much that his canine or molar teeth appeared.

Answer 1

A state of purity in this context refers to the time when a women is free from menstruation.

The hadith mentions that all these three men were with this woman during the same month of purity. The reason this caused an argument is because menstruation is a sign that someone is not pregnant with any child.

If they had been with the woman in different states of purity divided by menstruation, it would be easy to tell whose child it was. But, they were all with her in that one month, so it was unclear whose child it was.

There are many laws in Islam supposed to prevent this from happening including the waiting period for a woman that gets divorced or her husband dies. It is all to make sure menstruation occurs thus proving that the woman is not pregnant so that there is no doubt left whose child it is if she has a child with someone else after marrying them.

These men fell into this situation due to something unusual, but we are not told what it is. Perhaps this happened before they accepted Islam, then they came to Ali (RA) to decide what to do with the child after they had accepted Islam.

And Allah knows best.