Comprehension

Mathematicians are assigned a number called Erdős number (named after the famous mathematician, Paul Erdős). Only Paul Erdős himself has an Erdős number of \(0\). Any mathematician who has written a research paper with Erdős has an Erdős number of \(1\). For other mathematicians, the calculation of his/her Erdős number is as follows:

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

In a seven-day long mini-conference organized in memory of Paul Erdős, 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 with an Erdős number of \(\infty\).
  • Nobody had an Erdős number less than that of \(F\).

Event Timeline

Day 3 Event:

  • \(F\) co-authored a paper jointly with \(A\) and \(C\).
  • This reduced the average Erdős number of the group of eight mathematicians to \(3\).
  • The Erdős numbers of \(B, D, E, G, H\) remained unchanged.
  • No other co-authorship among any three members would have reduced the average Erdős number of the group to as low as \(3\).

At the end of Day 3:

  • Five members of the group had identical Erdős numbers.
  • The other three had Erdős numbers distinct from each other.

 

Day 5 Event:

  • \(E\) co-authored a paper with \(F\).
  • This reduced the group’s average Erdős number by \(0.5\).
  • The Erdős numbers of the other six members were unchanged.

Note: No other paper was written during the conference.

Question: 1

How many participants in the conference did not change their Erd\H{o}s number during the conference?

Show Hint

Track each participant across events — those unaffected in all events retain their original value.
Updated On: Jul 31, 2025
  • 2
  • 3
  • 4
  • 5
  • Cannot be determined
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The Correct Option is C

Solution and Explanation

From the conditions: - On day 3, F’s collaboration with A and C reduced the average to 3, changing Erd\H{o}s numbers of A and C.
- B, D, E, G, H remained unchanged at that point.
- On day 5, E collaborated with F, reducing the average further; E and F changed again, while six others stayed unchanged at that point.
- Overall, B, D, G, H never changed from start to end.
Thus, \( \boxed{4} \) participants did not change their Erd\H{o}s number.
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Question: 2

The person having the largest Erd\H{os number at the end of the conference must have had Erd\H{o}s number (at that time): }

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Infinity in such problems is reduced to a large finite number once connected; estimate using average constraints.
Updated On: Jul 31, 2025
  • 5
  • 7
  • 9
  • 14
  • 15
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The Correct Option is D

Solution and Explanation

Initially A had infinite Erd\H{o}s number, others finite. Through collaborations, the infinity dropped but remained the highest at the end. From given changes and average shifts, calculation shows the maximum finite number achievable here is \( \boxed{14} \).
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Question: 3

How many participants had the same Erd\H{os number at the beginning of the conference? }

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Equal Erd\H{o}s numbers occur when participants share the same closest linked co-author path to Erd\H{o}s.
Updated On: Jul 31, 2025
  • 2
  • 3
  • 4
  • 5
  • Cannot be determined
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The Correct Option is B

Solution and Explanation

From initial setup: - Only A was infinite, F was highest finite, others had lower. - Three participants shared the same starting number due to equal shortest-path connection to Erd\H{o}s. Thus, \( \boxed{3} \) participants had identical numbers initially.
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Question: 4

The Erd\H{os number of C at the end of the conference was: }

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Track each individual’s change points; final value is from last connected update.
Updated On: Jul 31, 2025
  • 1
  • 2
  • 3
  • 4
  • 5
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The Correct Option is C

Solution and Explanation

On day 3, C connected to F (who was connected closer to Erd\H{o}s), reducing C’s number. By final day, no further changes affected C. Thus, C’s final number was \( \boxed{3} \).
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Question: 5

The Erd\H{os number of E at the beginning of the conference was: }

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Identify initial values from network distances before any updates occur.
Updated On: Jul 31, 2025
  • 2
  • 5
  • 6
  • 7
  • 8
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The Correct Option is B

Solution and Explanation

From initial setup and average constraints, E was not directly connected to low-number participants, giving him a starting number of \( \boxed{5} \).
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