Sport climbing at the 2020 Summer Olympics – Women's combined

Warmup

Sport climbing is a new event (one of four) at the 2020 Summer Olympics in Tokyo, and it just finished. Two events were held, one for men and one for women. The format consists of three disciplines: lead climbing, speed climbing and bouldering. The winner is determined based on who performs best in all three disciplines.

Let's look at the women's event. In the final round, each of eight competitors competed in each of the three disciplines. Each competitor received a rank, a distinct integer from 1 to 8, in each of the disciplines. (There were no ties in the individual disciplines.) The scores were multiplied and sorted; the lowest scores received medals.

The final scores of the women's final round of sport climbing at the 2020 summer Olympics were as follows:

1. 🥇 Janja Garnbret: 5
2. 🥈 Miho Nonaka: 45
3. 🥉 Akiyo Noguchi: 64
4. Aleksandra Mirosław: 64
6. Anouck Jaubert: 84
7. Jessica Pilz: 90
8. Seo Chae-hyun: 112

As you can see, there were ties after multiplying the competitors' scores. Because these ties were both head-to-head (between two competitors), they were resolved by comparing the relative performance of the tied competitors in the three disciplines: whoever was better in more of the disciplines is ahead in the final order. (So, for example, we know that Brooke Raboutou performed better than Anouck Jaubert in more individual disciplines.)

Part one: Who were the winners of the three individual disciplines? (Of course, it's not possible to distinguish the disciplines from the information provided – how could you tell the difference between, eg, lead and bouldering? – but it's possible to say who the winners were.)

Full puzzle

The speed discipline has a rather unusual format. The competitors are seeded from 1 to 8 based on an earlier round. Then, in a standard bracket pairing (1 vs 8, 2 vs 7, 3 vs 6, 4 vs 5), the competitors compete in pairs for who can climb a standardized speed wall faster. The winners proceed to a winners bracket, while the losers proceed to a losers bracket. Everyone continues to participate in two more speed climbs; for example, one of the pairs in the second part of the bracket is (the winner of 1 vs 8) vs (the winner of 4 vs 5). After three such climbs, the contestants end up "sorted" and receive a corresponding rank.

It's honestly very hard to describe this process in words. Here's a picture of the bracket used: As you can see in the image above, the seeds for the speed round were as follows: 1. Aleksandra Mirosław, 2. Anouck Jaubert, 3. Miho Nonaka, 4. Akiyo Noguchi, 5. Jessica Pilz, 6. Brooke Raboutou, 7. Janja Garnbret, 8. Seo Chae-hyun.

The resulting sorted order is: 1. Winner of Race 12, 2. Loser of Race 12, 3. Winner of Race 11, 4. Loser of Race 11, 5. Winner of Race 10, 6. Loser of Race 10, 7. Winner of Race 9, 8. Loser of Race 9. These ranks are the ones used in the multiplication of the final scores above.

Bouldering and lead use a more straightforward format, which I will not describe here because the information does not help reconstruct the results. For both the bouldering and lead disciplines, the competitors climb and end up with ranks from 1 to 8 in a relatively direct manner.

Necessary hint for the puzzle below: I personally find the lead discipline rather enjoyable to watch, as I enjoy lead climbing myself. It's nice that the winner of the lead discipline did so well overall, but I was sad that the 2nd and 3rd places in lead (who, presumably, would have won the silver and bronze medals had there been separate medals for lead climbing) ended up in the last two places (7th and 8th) of the combined ranking!

Part two: What were the full sorted orders of the competitors in all three disciplines?

• Go Janja! I had no doubts that she will win - she was so much better than everyone. Aug 7, 2021 at 9:24
• Thanks, this was a fun puzzle! Aug 8, 2021 at 13:56

Part One:

Starting from the beginning:

We can unambiguously determine the three factors used in 1st, 2nd and 7th places; they must be 1x1x5, 3x3x5, and 3x6x5 respectively, since no individual placement can be greater than 8.

Then, onto 5th and 6th places

breaking 84 down into prime factors, we have 2x2x3x7. Since our 2nd and 7th places must already account for all three 3rd places, they must both be 2x6x7.

Onto 2nd and 3rd:

64 = 2^6 and so we have 2x4x8, 4x4x4 or 1x8x8 as possible answers. 4x4x4 must combine with 1x8x8, and since we have already used two of our 2nd places, they also cannot both be 2x4x8; therefore, at least one of them is 1x8x8 and is our third 1st placer. We cannot at this time unambiguously determine the placements of the other two; they are either 2x4x8 + 4x7x4 or 4x4x4 + 2x7x8.

Finally

1x8x8 always loses the tiebreak, since in the 2x4x8 case the 1st place must necessarily line up with the other contestant's 8th.

Thus:

Janja Garnbret won two disciplines, and Aleksandra Miroslaw won the other.

On to Part Two:

The hint resolves the ambiguity and

puts Seo on a 2nd place, making her placements 2nd, 7th and 8th. Seo also did not get 8th in lead (she got 2nd), therefore, Miroslaw (who placed 8th in 2 disciplines) did, putting Garnbret in 1st on the discipline. Raboutou and Jaubert must have gotten 6th and 7th. Since Raboutou outranks in the final standings with the same set of results, she must have placed 6th here (to be able to win the tiebreaker 2-1). Once again, our hint resolves the last ambiguity; it places Pilz in 3rd in Lead, and thus Nonaka must be 5th.

Garnbret, Seo, Pilz, Noguchi, Nonaka, Raboutou, Jaubert, Miroslaw

On to the Speed bracket:

Seo placed either 7th or 8th; this means her opponent in the first round, Miroslaw, must have placed in the top half of the bracket. Since her total placements are 1x8x8, she must have placed first in the final bracket, and since Garnbret only has one result not in 1st place, she in turn must have placed 5th. In turn, overall 2rd place Nonaka (3x3x5) must have placed 3rd in Speed. This puts Raboutou in the bottom half of the bracket, but more importantly, puts Pilz (3x5x6) in 6th by elimination - which in turn puts Raboutou (2x6x7, 6th in Lead) in 7th. Jaubert (also 2x6x7) must be 2nd, leaving Seo in 8th.

Speed results:

Miroslaw, Jaubert, Nonaka, Noguchi, Garnbret, Pilz, Raboutou, Seo

By elimination, Bouldering results must be

Garnbret, Raboutou, Nonaka, Noguchi, Pilz, Jaubert, Seo, Miroslaw.

• Nice job! I hope you enjoyed solving the puzzle. I personally like how natural part one is, and that it involves reasoning about the tiebreak. (One minor comment on your writeup about something that is hard to know except by eagle-eyed examination of the bracket: Seo is the family name of Seo Chae-hyun. All the other climbers have the family name you expect.) Aug 7, 2021 at 0:17
• Korean name; that detail slipped my mind entirely. Edited. Part one is certainly a lot cleaner than the full version; I hung up on the tiebreak for about a minute before realizing that obviously, one discipline does not have two people placing 8th. Aug 7, 2021 at 0:28