Erdos Number ( from CAT 2006)
Mathematicians are assigned a number called Erdos number(named after the famous mathematician, Paul erdos).
Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of 1. For other mathematicians, the calculation of his Erdos 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 Erdos number. Let the Erdos number of Y be y. Then X has an Erdos number of y+1.
Hence any mathematican with no co-authorship chain connected to Erdos has an Erdos number of 8(infinity).
In a seven day long mini-conferenceorganized in the memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problem. At the begining of the conference, A was the only participant who had an infinite Erdos number. Nobody had an Erdos 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 Erdos number of the group of eight mathematicians to 3. The Erdos numbers of B, D, E, G and H remained unchanged with the writing of the paper. Further, no other co-authorship among three members could have reduced the average Erdos 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 Erdos numbers while the other three had Erdos numbers distinct from each other.
On the fifth day, E co-authored a paper with F. This reduced the groups average Erdos number by 0.5. The Erdos number of the remaning six were unchanged with the writing of this paper.
No other paper was written during the conference.
(i) The Erdos number of C at the end of the conference was
(a) 1 (b) 2 (c) 3 (d) 4 (e) 5
(ii) How many participants had the same Erdos number at the begining of the conference?
(a) 2 (b) 3 (c) 4 (d) 5 (e) Cannot be determined
(iii) The Erdos number of E at the begining of the conference was
(a) 2 (b) 5 (3) 6 (d) 7 (e) 8
(iv) How many participants in the conference did not change their Erdos number during the conference?
(a) 2 (b) 3 (3) 4
(d) 5 (e) Cannot be determined
(v)
The person having the largest Erdos number at the end of the conference must
have had Erdos number (at that time): (a) 5 (b) 7 (3) 9
(d) 14 (e) 15
Solution