# Help me settle a bet between 2 friends [closed]

They were driving go carts a few months ago and now the topic came up that friend1 (who was first in race) overtook friend2 (who was second in the race), of course friend1 claims he overtook him and friend2 claims he didnt.

friend1 = { 57.03, 55.50, 55.39, 55.62, 55.26, 55.79, 55.15, 55.69, 58.28, 54.72, 56.02, 54.61, 58.27, 55.19, 54.70, 54.93, 56.31 };
friend2 = { 59.98, 59.07, 58.84, 58.24, 58.24, 58.41, 58.79, 58.02, 59.20, 58.09, 57.60, 57.99, 59.10, 58.29, 58.21, 57.49, 60.04 }


Those are the lap times, friend1 claims he overtook friend2 at around lap 12-13. Is there any way to calculate this? Does the lenght of the track matter? Also friend1 has one more lap clocked at the end but because he said he overtook him earlier and we dont know exactly if other drivers were clocked after that still or not i dont think it matters (correct me if i am wrong).

So did friend1 overtake friend2 at any point in the race?

No one was overtaken during this race.

That is, assuming they started at the same point and time and the first lap mentioned is also the first lap of the race.

Calculation for this:

friend1 completed the 17 laps in 948.46 seconds, whereas friend2 finished his 16th lap at 935.56 seconds, so at that time friend2 wasn't overtaken by friend1.
friend1 was faster than friend2 in every lap, so if he wasn't more than a lap ahead of him at the very end, he cannot have been more than a lap ahead of him at the end of any lap.
Theoretically, as elias correctly pointed out in the comments, it is possible that friend1 overtook friend2 during a lap and, before the end of that lap, friend2 overtook friend1 again. However, the difference between friend1 and friend2 has consistently been more than 10 seconds at the end of each lap, so this is very unlikely. Besides, the lap times are very stable throughout, which makes this scenario even more unlikely.

• I don't see your last conclusion supported. Theoretically it is possible, that Friend1 did the majority of his last lap - for the sake of argument - in 10 seconds, overtaking Friend2 and stopped just before the finish line to celebrate. During the celebration Friend2 did reovertake him (to still be 1 lap back) and finished his own 16th lap. Later Friend1 finished his 17th lap. Of course this is an artificial example, but the numbers do not prove there was no overtake at all. – elias Feb 16 '17 at 12:23
• @elias: Hah, yes, didn't think of that one, will edit it into my answer. – Levieux Feb 16 '17 at 12:25
• I don't see how friend1 is one lap ahead. He is close to lapping friend2 at the end of the race(if we take the 17th lap timings as well, he is 47.2 seconds ahead, which is just short of one lap by his timings) – Sid Feb 16 '17 at 12:44
• @Sid, my point was that it is impossible to tell if there was an overtake or not. It cannot be proven if there was one, just as it cannot be proven there wasn't. The given numbers just don't provide enough information to decide that. – elias Feb 16 '17 at 12:49
• Levieux showed a scenario in which there was no overlap, and I showed one in which there was, both satisfying the numbers. – elias Feb 16 '17 at 12:59