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Suppose 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, E, G 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.

Q 1. The person having the largest Erdös number at the end of the conference must have had Erdös number (at that time): -
(1) 5
(2) 7
(3) 9
(4) 14
(5) 15

Q 2. How many participants in the conference did not change their Erdös number during the conference?
(1) 2
(2) 3
(3) 4
(4) 5
(5) Cannot be determined

Q 3. The Erdös number of C at the end of the conference was: -
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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