# Staggered Race Puzzle

This is puzzle on staggered race

This is the Solution is given for question in image

Question : What could be the maximum time for which Neeraj could have been in the second position in the race?

Solution given for this is here:

I am facing issues in visualization of the puzzle. Pls help.

Source - TIME

• Provide attribution; i.e., where and how did you come across this puzzle? – user477343 Sep 26 '18 at 12:45
• Just as a suggestion, when you give the solution of a puzzle you posted, please put it in spoilers so that other people can give it a go first :D – Kevin L Sep 27 '18 at 2:44

27 seconds.

Because:

Because one person can no longer overtake another if the other has already overtaken him, Neeraj can only be second place during the lead of two consecutive people. Why two? Let's assume he started at position 2. For 13 seconds, no overtakings take place. Then, former position 3 (Harsh) overtakes Neeraj and Ashish, while Ashish falls behind you. Neeraj is now still position 2 for the next period (14 seconds). But now Neeraj can't stay on position 2 because he cannot overtake Harsh anymore, while Devesh overtakes them both. He will never get to position 2 any more. The longest period of two consecutive leaders is 27 seconds (Ashish and Harsh), and that will be my answer.

• Answer to his question is given as 27 seconds. I will add the image of the solution given. I am also having a hard time understanding their reasoning. Pls help. – sam Sep 26 '18 at 14:32
• @sam oh yes 27 is correct actually. I have tried to explain now – Kaspar Scherrer Sep 26 '18 at 14:59
• But if Neeraj was second while Ashish and Harsh are in the lead, how can he then still be second at the end of the race? Instead he must be second while Karan and Ishant are in the lead, which also happens to be 27 seconds. – Jaap Scherphuis Sep 26 '18 at 15:10
• @JaapScherphuis nobody said anything about being second at the end of the race (That is a condition for the other question, which the answers shouldn't focus on). And no, Karan and Ishant are in the lead together for 17 seconds ;) – Kaspar Scherrer Sep 26 '18 at 15:11
• @Cashbee: Oh right. Neeraj being second at the end was only part of the first question. And I also can't count. – Jaap Scherphuis Sep 26 '18 at 15:14

Why is it 27 seconds?

The situation is now unlinked from him finishing second at the end - or where he started at the beginning (we can choose his start and end position to our advantage).

Because you can only overtake or be overtaken by each person he can be in second position exactly twice. By being in second place (overtaking the current second place) and again by overtaking the leader - whilst simultaneously being overtaken by someone else

(a third time is impossible as he cannot now overtake the new leader)

So we are looking for two adjacent leaders times that add to the most - which is the first two at 27 seconds.

this leaves him finishing in 9th place

• He can never overtake the leader, or else he'd show up in the time table. But overtaking the leader is not needed. The third can overtake Neeraj, then the leader, then Neeraj overtakes the former leader. Therefore Neeraj's second place is interrupted for an instance while he is third. This instance could be next to no time if they are all very close together, so saying $27$ should be close enough. – Kaspar Scherrer Sep 26 '18 at 15:09
• @cashbee thats just poor english on my part - I'll edit to make clearer – Collett89 Sep 27 '18 at 6:44

So we have 10 runners of which Neeraj is 1

So a simple upper bound is 9

Think of a typical 400m footrace and how the inner lanes start further back - because they run a shorter route on the bends - so the inside lane starts "behind"

If Neeraj starts in last (the inside lane) and runs quicker than everyone but Ishant (who must have started 9th) Then he would finish 2nd.

Meaning

And to answer about how long he could have been in second place

Neeraj started behind Ishant and at no point overtakes him - meaning he did not overtake Karan before Ishant did - thus the soonest he could have done so is 1:21 - meaning 9 seconds between then and completion

thus

9 seconds assuming Ishant and Neeraj overtook karan simultaneously (negligible time difference between them)

A little caveat

if Neeraj starts infront of Ishant and we assume that Ishant overtakes Neeraj as the pair overtake Karan - then Neeraj is in second from 1:13 onwards - a total of 17 seconds

so

17 seconds if Neeraj starts infront of Ishant

• OP asked his own question. don't answer the question in the image ;) (I think?) – Kaspar Scherrer Sep 26 '18 at 13:10
• My doubt is the question below the image... – sam Sep 26 '18 at 13:11
• @Collet89 i also feel the answer should be 9 but unfortunately it is 8 – sam Sep 26 '18 at 13:11
• @cashbee - noted - i'll add a fix for that now – Collett89 Sep 26 '18 at 13:12
• @sam secondary question answered - id like to see the logic for 8 - it doesnt mention consistent running speeds so the simple solution seems impossible to disprove – Collett89 Sep 26 '18 at 13:25