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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös number is illustrated below:

Suppose that a mathematician X has co-authored papers with several other mathematicians. From among them, mathematician Y has the smallest Erdös number. Let the Erdös number of Y be *y*. Then X has an Erdös number of *y* + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity.

In a seven day long mini-conference organized in memory of Paul Erdös, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems. At the beginning of the conference, A was the only participant who had an infinite Erdös number. Nobody had an Erdös number less than that of F.

• On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös number of the group of eight to as low as 3.

• At the end of the third day, five members of this group had identical Erdös numbers while the other three had Erdös numbers distinct from each other.

• On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös numbers of the remaining six were unchanged with the writing of this paper.

• No other paper was written during the conference.

How many participants had the same Erdös number at the beginning of the conference?