National Games event puzzle

[unrealdon]

There is a decathlon event in national games where 131 people are participating.

One athlete manages to gain 114th position in every individual event.

The puzzle is to find out the highest place he can finish in overall decathlon event?

[bonanova]

104

[fabpig]

on the scoring system.

If it's similar to the Olympic decathlon where points are awareded for performance rather than psition in the indvidual events, it is theoretically possible for the competitor to win

[TheChad08]

on the scoring system.

If it's similar to the Olympic decathlon where points are awareded for performance rather than psition in the indvidual events, it is theoretically possible for the competitor to win

I would assume that the scoring system is based upon position and differs by 1 per spot.

Whether you want to award 131 points for first and 1 point for last and whoever has the most points wins.

OR

Probably easier would be to give first place 1 point and 131st 131 points and make it like golf where the lowest score is best.

[bonanova]

104

97

To begin, let the 131 contestants C_i achieve i^th place in every event, respectively accruing total placement points of 10, 20, ..., 1140, ..., 1300, 1310. To optimize the standing of contestant C₁₁₄ we seek to create a maximum of contestants with 1141 or more placement points. This maximizes the number of contestants that place behind the 1140 placement points of C₁₁₄.

We alter placement point totals by swapping pairs of contestants within an event. For example, if C₁₁₃ and C₁₂₄ swap places in Event 1, their points will change by 11: from 1130 and 1240 to 1141 and 1229 respectively. This single swap moves C₁₁₄ up one position, and of course further swaps can move him even farther along.

How far can we push C₁₁₄ up in the standings?

Consider the 17 contestants C₁₁₅ - C₁₃₁ that initially place behind C₁₁₄. Their combined points, 1150 + 1160 + ... + 1300 + 1310, is 20910, which is 1513 greater than their combined total would be if they all had 1141. So 1513 of their points can be added to the points of C₁₁₃, C₁₁₂, C₁₁₁, ... without placing any of them ahead of C_114.

How many contestants with initially fewer points can have their points increased to 1141 or greater? The contestants who immediately precede C₁₁₄ must have 11, 21, 31, 41, ... added to their respective totals. The partial sum of that sequence reaches 1506 [OK] at C97 and 1687 [more than 1513] at C₉₆.

Thus C₁₁₄ can achieve 97^th place.

This is an upper limit. I have not specified the complete set of swaps; however there seems to be enough flexibility to achieve it.

[unrealdon]

you can take the scoring to be exactly like that in Olympics and I dont see any way one could win if he finishes 114th in all events.

bonanova is close.

[unrealdon]

I would assume that the scoring system is based upon position and differs by 1 per spot.

Whether you want to award 131 points for first and 1 point for last and whoever has the most points wins.

OR

Probably easier would be to give first place 1 point and 131st 131 points and make it like golf where the lowest score is best.

First assumption holds good for scoring.

[fabpig]

you can take the scoring to be exactly like that in Olympics and I dont see any way one could win if he finishes 114th in all events.

In event 1 (and in all events), our competitor finishes 114th with a score of 1000 points. Those ahead of him, pos 1-113 score 1001 points each (just for ease of maths).

In event 2, those that finished 1-17 in event 1 are disqualified and fill positions 115-131 with 0 points. All competitors move up 17 places from event 1

And so on.

After just 8 events, all the other 130 competitors (<( 8*17)) have had at least 1 event with 0 points. Our man has 8000 points, nearest competitor has 7007 points.

And he's won at a canter. Dunno why the others turned up.

Yeah, it's never going to happen. Be nice if it did tho.

[unrealdon]

In event 1 (and in all events), our competitor finishes 114th with a score of 1000 points. Those ahead of him, pos 1-113 score 1001 points each (just for ease of maths).

In event 2, those that finished 1-17 in event 1 are disqualified and fill positions 115-131 with 0 points. All competitors move up 17 places from event 1

And so on.

After just 8 events, all the other 130 competitors (<( 8*17)) have had at least 1 event with 0 points. Our man has 8000 points, nearest competitor has 7007 points.

And he's won at a canter. Dunno why the others turned up.

Yeah, it's never going to happen. Be nice if it did tho.

LoL How do you think all the 113 people can have same score and still have different ranks. Won't they all be having first rank if they had same score?

[fabpig]

LoL How do you think all the 113 people can have same score and still have different ranks. Won't they all be having first rank if they had same score?

But the guy following would be 114th as he had 113 people ahead of him. But I just used those figures for ease. Working on a similar principle, you could have 1st on 1113 points, 2nd on 1112, 3rd on 1111......113th on 1001 pts. There would still be enough points to win comfortably.

Edit....spelling

Edited August 6, 2012 by fabpig

[unrealdon]

But the guy following would be 114th as he had 113 people ahead of him. But I just used those figures for ease. Working on a similar principle, you could have 1st on 1113 points, 2nd on 1112, 3rd on 1111......113th on 1001 pts. There would still be enough points to win comfortably.

Edit....spelling

For the sake of solution of puzzle, lets assume first person wins 131 points, second 130 and so on ... last person gets 0 points, all seperated by one point.

Edited August 7, 2012 by unrealdon

[bonanova]

you can take the scoring to be exactly like that in Olympics and I dont see any way one could win if he finishes 114th in all events.

bonanova is close.

I guess you could juggle the points among leading contestants and push one or two more down below 1140. I had the feeling that starting out with C_i placing i^th would have the effect of top loading the best finishers thereby pushing the others down maximally, but I didn't check that option out. We have 7 points from the trailing contestants to play with, but it takes 186 points to push C₉₆ down. Someone want to find those 179 other points, have at it.

It's an interesting puzzle because there are too many contestants to just use pencil and paper, but the solution lies in a small enough corner of the search space that monte carlo methods are useless as well. Tip of the hat.