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Part I

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, 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.


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

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

3. The Erdös number of C at the end of the conference was:

(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

4. The Erdös number of E at the beginning of the conference was:

(1) 2
(2) 5
(3) 6
(4) 7
(5) 8

5. How many participants had the same Erdös number at the beginning of the

conference?
(1) 2
(2) 3
(3) 4
(4) 5
(5) Cannot be determined

Answers-

1. (2) 

2. (4)

3. (2)

4. (3)

5.(2)

Part II

K, L, M, N, P, Q, R, S, U and W are the only ten members in a department. There is a proposal to form a team from within the members of the department, subject to the following conditions:


A team must include exactly one among P, R, and S.
A team must include either M or Q, but not both.
If a team includes K, then it must also include L, and vice versa.
If a team includes one among S, U, and W, then it must also include the other two.
L and N cannot be members of the same team.
L and U cannot be members of the same team.
The size of a team is defined as the number of members in the team.

1. What could be the size of a team that includes K?

(1) 2 or 3
(2) 2 or 4
(3) 3 or 4
(4) Only 2
(5) Only 4

2. In how many ways a team can be constituted so that the team includes N?

(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

3. What would be the size of the largest possible team?

(1) 8
(2) 7
(3) 6
(4) 5
(5) Cannot be determined

4. Who can be a member of a team of size 5?

(1) K
(2) L
(3) M
(4) P

(5) R

5. Who cannot be a member of a team of size 3?

(1) L
(2) M
(3) N
(4) P
(5) Q

Answers-

1. (5)

2. (5)

3. (4)

4. (3)

5. (1)

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