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Seven athletes — A1, A2, A3, A4, A5, A6, and A7 participated in a 100 m race on each day of a particular week from Sunday to Saturday. Ranks, from 1 to 7, were given to the athletes after each race. Rank 1 was given to the athlete who was the fastest to complete the race, Rank 2 was given to the athlete who was the second fastest to complete the race, and similarly Rank 7 was given to the athlete who was the slowest to complete the race. The table provides information about the ranks received by the athletes in the races on Sunday, Monday, Tuesday, Wednesday, Thursday, and Friday.
One athlete is said to be better than another athlete, if he/she has received a better Rank than another athlete in at least 4 races.

_________|Rank 1|Rank 2|Rank 3|Rank 4|Rank 5|Rank 6|Rank 7|
Sunday | A4 | A7 | A2 | A5 | A6 | A3 | A1 |
Monday | A3 | A6 | A5 | A2 | A4 | A7 | A1 |
Tuesday | A1 | A6 | A2 | A4 | A3 | A5 | A7 |
Wednesday| A5 | A3 | A7 | A1 | A4 | A2 | A6 |
Thursday | A2 | A5 | A7 | A6 | A1 | A4 | A3 |
Friday | A6 | A7 | A3 | A2 | A1 | A5 | A4 |

If A4 is not better than any of the given athletes, then how many different ranks could have been received by him/her in the race on Saturday?

A3 could have been better than at most how many athletes?

Please tell how to approach this.

Source : IMS

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  • $\begingroup$ Why apologize for an image question when you can type it out? :) $\endgroup$
    – Chowzen
    Oct 13, 2018 at 12:11
  • $\begingroup$ @Chowzen how does the formatting look on the website? It seems to be off a bit on mobile, not sure if it’s better on the website. $\endgroup$
    – El-Guest
    Oct 13, 2018 at 12:15
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    $\begingroup$ @El-Guest Like this. And the original pic is linked/preserved. $\endgroup$
    – Chowzen
    Oct 13, 2018 at 12:23
  • $\begingroup$ @Chowzen Yeah, I think the mobile formatting is hopelessly messy, but your edit looks good!! Thanks for doing that! :D $\endgroup$
    – El-Guest
    Oct 13, 2018 at 12:25

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We’ll start with A4:

Let’s see how many people A4 has beaten so far: they’ve beaten A1 twice, A2 twice, A3 three times, A5 twice, A6 twice, and A7 three times. In order for them not to be better than A3 or A7, A4 must lose to both of them on Saturday. A4 may therefore finish no higher than 3rd, and so they may hold 5 different possible ranks on Saturday (3rd, 4th, 5th, 6th, and 7th).

Now let’s consider A3:

To maximize our number of athletes who are worse than A3, let’s assume A3 wins on Saturday. If this is the case, A3 has beaten A1 five times, A2 four times, A4 four times, A5 four times, A6 three times, and A7 four times. A3 can at most be better than 5 other athletes (A1,A2,A4,A5,A7).

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