Climbing complexity
In the 2020 Olympic Games, sport climbing was introduced for the first time, and something very interesting happened with the scoring system. Can you find out what was interesting about it?
Problem
In the 2020 Olympics, sport climbing was introduced for the first time. There were three different climbing disciplines combined under one medal:
- Speed climbing
- Bouldering
- Lead climbing
For each of these three disciplines, the eight athletes in the final were given a score from 1 to 8, with the athlete who performed the best getting a score of 1. The three scores were then multiplied together at the end. For example, if a climber came 2nd in lead climbing, 3rd in bouldering and 8th in speed climbing, their score would be $2×3×8=48$. The person with the lowest score overall was awarded the gold medal.
In the men's final, just before Jakob Schubert climbed the final lead climb of the competition, this is what the scoreboard looked like:
| Speed climbing | Bouldering | Lead climbing | |
|---|---|---|---|
| Alberto Lopez | 1 | 7 | 3 |
| Adam Ondra | 4 | 6 | 1 |
| Nathaniel Coleman | 6 | 1 | 4 |
| Tomoa Narasaki | 2 | 3 | 5 |
| Jakob Schubert | 7 | 5 | |
| Mickaël Mawem | 3 | 2 | 6 |
| Colin Duffy | 5 | 4 | 2 |
| Bassa Mawem | 8 | 8 | 7 |
(Bassa Mawem was injured and did not compete.)
Just before Schubert's climb, the commentators wanted to work out who might get medals, depending on how well Schubert climbed.
Imagine you are a commentator for this competition. Create a table of all the possible final scores, depending on the score Schubert gets for his final climb.
Does anything surprise you about your table?
Extension:
How else could the scoring have been done? Think of a scoring method of your own, and then investigate how this method would change the possible final scores.
What are the advantages and disadvantages of your method? What are the advantages and disadvantages of the method that was actually used in the 2020 Olympics?
Speed climbing is very different to the other two types of climbing, so athletes who are the best in the world at speed climbing might not necessarily do very well at bouldering or lead climbing. But the organisers wanted to make sure that the best speed climbers in the world would still have a good chance of getting a medal. How might this have affected their choice of scoring method?
Teachers' Resources
Why do this problem?
This problem provides an engaging context for exploring multiplication. The fact that one athlete's score in the lead climbing can drastically change the positions of the other climbers on the leaderboard will spark learners' curiosity, and this will encourage them to notice facts such as how whoever comes first on a wall is at a huge advantage due to the nature of multiplying by 1. There are other facts about multiplication to be explored here, such as that going from a score of 1 to 2 will double an athlete's overall score, whereas going from a score of 5 to 6 will only multiply their overall score by 6/5.
Possible approach
You might like to spend some time as a whole class calculating what would happen if Schubert came eighth, and then discussing what students notice about those initial scores. Some students will notice that the top three athletes all came first in one of their three climbs, and that there is a large gap between the best six scores (ranging from 21 to 40) and the bottom two scores (280 and 448).
Make sure learners know that as Schubert moves up the rankings, everybody else's scores for the lead climbing will drop down, one by one. Once everybody is confident with how the scoring works, allow some time for students to create their tables and investigate how the scores would change as Schubert progresses through his climb. You might like to bring the class back together at various points to discuss different learners' ideas and to draw attention to anything interesting that students have noticed.
At the end of the activity, bring the whole class back together to discuss the problem. Why might this scoring system have been used? What are the pros and cons of it? You might then like to tell them that in the 2024 Olympics, there was a separate medal for speed climbing, and the combined scores for bouldering and lead climbing were calculated by adding the scores together rather than multiplying them. Why do students think the system might have been changed in this way?
Key questions
What are everybody's scores at this point? Who would get the gold, silver and bronze medals if this was the final outcome?
Is anybody guaranteed a medal no matter how well Schubert climbs?
What is the most interesting thing that happens in your table?
Is it best to be first in one discipline and then not do very well in the other two disciplines, or to do quite well in each? Why?
Possible extension
Students might like to have a go at modelling this situation with their own athletes - perhaps fewer than eight to make things simpler. They can decide on the order of their athletes for the speed climbing and the bouldering events, and then see what happens as they go through the lead climbing. What is the strangest thing they can make happen on the leaderboard?
Possible support
Climbing Conundrum is a version of this task with fewer climbers, which might be a good starting point to access this task from. Some learners will also benefit from having a calculator available to work out the scores.
Additional information:
We have specifically focused on the men's final in this activity because the women’s 2020 final was much less close. Janja Garnbret climbed exceptionally well and was guaranteed the gold medal before the final two athletes climbed, as their scores were too large to beat hers even if they had managed to climb all the way to the top.
In the 2024 Olympics, the scoring system changed. There were two separate sets of medals awarded – one for speed climbing, and one for lead climbing and bouldering combined. For the lead and bouldering medal, climbers’ scores for each of the two disciplines were added together rather than multiplied. Take a look at the Sport Climbing at the Summer Olympics Wikipedia page for more information.
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