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?
Student Solutions
Well done to everybody who explored what was happening with this unusual scoring system. Some solutions didn't take into account that as Schubert overtook other athletes on the final wall, the other athletes' scores on that wall would also change. We'll focus here on the solutions which did take that into account.
Max from Stanborough WGC in England worked out where Schubert's final place in the leaderboard could be:
Jakob can only be 3rd or 2nd to bottom.
Best= 3rd
2nd+=2nd to last
This is interesting, Max - normally we would expect an athlete to gradually rise through the rankings as their score improved, but it does look like overall Jakob Schubert will either come 7th or 2nd, depending on whether or not he beats all of the other athletes on the final wall.
Dhruv from The Glasgow Academy in Scotland looked at where some of the other athletes might place, depending on Schubert's score:
If Jakob comes 2nd or 3rd in LC (lead climbing) that means Adam Ondra wins otherwise Alberto Lopez wins if Jakob comes in any other position in LC.
Well spotted, Dhruv. From the way the leaderboard changes as Schubert climbs, it looks like Lopez and Ondra will switch places a couple of times depending on how far he gets!
Dhruv drew tables of all the possible outcomes, including this table, which shows what would happen if Schubert came first in the lead climbing:
| Name | SC | B | LC | Total |
| Alberto Lopez | 1 | 7 | 4 | 1*7*4=28 |
| Adam Ondra | 4 | 6 | 2 | 4*6*2=48 |
| Nathaniel Coleman | 6 | 1 | 5 | 6*1*5=30 |
| Tomoa Narasaki | 2 | 3 | 6 | 2*3*6=36 |
| Jakob Schubert | 7 | 5 | 1 | 7*5*1=35 |
| Mickaël Mawem | 3 | 2 | 7 | 3*2*7=42 |
| Colin Duffy | 5 | 4 | 3 | 5*4*3=60 |
| Bassa Mawem | 8 | 8 | 8 | 8*8*8=512 |
Dhruv said:
The thing that surprises me is that the top 2 doesn't change whatever happens.
Have a look at the spreadsheet showing Dhruv's full solution. I think the table above shows that actually, Ondra isn't always guaranteed to place either first or second.
In the actual competition, Schubert did come first in the lead climbing. I wonder what Adam Ondra was thinking throughout Schubert's final climb!
Abdullah from Doha College in Qatar thought about the qualities of this scoring system, and came up with an alternative system:
Jakob Schubert’s position in Lead Climbing before his final climb determined the possible outcomes for his total score, ranging from 35 (if he placed 1st in Lead Climbing) to 280 (if he placed 8th). This system emphasizes the importance of consistency across all disciplines but is also sensitive to a single event’s performance, making it highly dynamic.
An alternative scoring method could involve weighting the disciplines differently, such as giving Speed Climbing 50%, Bouldering 30%, and Lead Climbing 20% in terms of importance. This would reduce the penalty for athletes who specialize in one event, making the scoring more reflective of their strengths. However, this could make the competition less fair to athletes who perform well across multiple disciplines. The original scoring method ensures a balance between specialization and overall performance, giving athletes a fair chance regardless of their focus in one particular discipline.
Your alternative scoring method is similar to what happened in the 2024 Olympics, Abdullah - there were two medals, one for speed climbing and one for bouldering and lead climbing combined, which is a bit like giving them ratings of 50%, 25% and 25% across the two medals.
Kwunting from Dulwich College in Beijing, China did a lot of thinking about the advantages and disadvantages of this scoring method, especially compared to an addition scoring method:
The biggest surprise is how much the multiplicative scoring system amplifies even a single poor result. Schubert needs to get first to have a chance at getting a medal, and it majorly punishes a competitor’s score even for a single error.
To fix this I believe a better Alternative Scoring Method is the Sum of Rank
Total Score = Speed Climbing Rank + Bouldering Rank + Lead Climbing rank
Benefit of New Addition Method:
1. Reduces the Impact of One Bad Result
Multiplication punishes a single bad rank (example: a score of 7 or 8 multiplies everything). While using addition, a single poor event doesn’t completely destroy your medal chances.
For example, under multiplication:
o A rank of 1 × 2 × 8 = 16
o A rank of 2 × 2 × 8 = 32
o Huge difference for a small change!
In comparison, under addition:
o 1 + 2 + 8 = 11
o 2 + 2 + 8 = 12
o Much Smaller penalty.
2. Fairer to Athletes specialising in a single area
• An athlete who is very good at one area but average in others can still place well.
• With multiplication, you have to be excellent in all three areas
• Addition gives athletes room to still win medals without being totally penalized for a 5th or 6th place.
Drawbacks of New Addition Method:
1. Doesn’t Reward Dominance Strongly Enough
• An athlete who dominates one or two area but places low in the third might not get rewarded fairly.
• Example:
o 1st, 1st, and 8th = 10 points
o 3rd, 3rd, and 3rd = 9 points
Despite two 1st-place finishes, the first athlete ranks lower.
2. More Ties
• Addition makes fewer unique scores (examples: many ways to total 12), especially in small competitions.
• This increases the need for tiebreakers, which adds complexity to results and might feel unfair to some athletes, also confusing the audience.
These are great points, Kwunting. We haven't given you enough data to explore this here, but another consideration is that you don't have to rank athletes on each wall. In the 2024 Olympics, an addition scoring method was used, but instead of athletes being ranked the points were given depending on factors such as which exact climbing hold an athlete climbed up to before they fell. This reduced the probability of ties happening at the end of the competition.
Kwunting's point about the multiplication scoring system rewarding dominance in one or two areas does seem to have been a major consideration of the organisers. Kwunting explains more about this:
2. Why Multiplication Helps Speed Climbers
If the organizers had used the addition method, speed climbers who placed near the bottom in bouldering and lead would end up with very high total ranks making it almost impossible to medal.
But with multiplication, if a speed climber gets:
• 1st in speed, and
• moderate ranks (say 5th or 6th) in the others,
they can still get a competitive product.
Example:
1st (Speed), 6th (Boulder), 5th (Lead) → 1×6×5=30. That might still be a medal-winning score!
So, the multiplication method gives speed specialists an encouragement
• They can’t just rely on speed alone,
• But they can still be in top three if they’re average in the other two.
In Summary
Because speed climbing is so different from the other two events, the organizers used a multiplicative scoring method to make sure that:
• All area mattered, and
• Speed climbers weren’t unfairly penalized just for being weaker in bouldering or lead.
This method gives an easier path to medals for specialists, while still encouraging all athletes to be well rounded.
Take a look at Kwunting's full solution to see more of their ideas.
Jex from Impington Village College in England had lots of observations about what might happen:
First I decided to total up the contestants' scores from the speed climbing and bouldering as they had already happened and could no longer be changed:
Lopez - 7
Ondra - 24
Coleman - 6
Narasaki - 6
Schubert - 35
M. Mawem - 6
Duffy - 20
B. Mawem - 64
As Bassa Mawem was injured and unable to compete, he must come last in all of the climbs, including in the lead climbing, giving him a score of 8×8×8=512.
Next I created a table showing the scores of the contestants depending on where Schubert came
1st. 2nd. 3rd. 4th. 5th. 6th. 7th.
Lopez. 28. 28. 28. 21. 21. 21. 21.
Ondra. 48. 24. 24. 24. 24. 24. 24.
Coleman. 30. 30. 30. 30. 24. 24. 24.
Narasaki. 36. 36. 36. 36. 36. 30. 30.
Schubert. 35. 70. 105. 140. 175. 210. 245.
M.Mawem. 42. 42. 42. 42. 42. 42. 36.
Duffy. 60. 60. 40. 40. 40. 40. 40.
B.Mawem. 512. 512. 512. 512. 512. 512. 512
Things I noticed:
Lopez is guaranteed a medal, and will get gold unless Schubert comes 2nd or third.
Ondra gets a medal unless Schubert comes first, and gets gold if Schubert comes 2nd or third, and gets silver if Schubert comes worse than third.
Coleman is guaranteed a medal, but (can't) get gold.
Narasaki gets a bronze medal if Schubert gets 5th or worse. (I think Jex is thinking that both athletes who tie for second place will get silver medals, and then the athlete in the next place will get bronze. In the Olympics, Narasaki would actually place fourth in this case and wouldn't get a medal.)
Schubert can only get a medal if he comes first: even at this point he only gets bronze.
M.Mawem, Duffy and B.Mawem can all never get medals.
Only Lopez or Ondra can come first.
Jex thought of some different scoring methods, and compared these against the multiplication method:
My scoring method(s)
1. Add all numbers together and then times by the smallest. The winner is the one with the smallest overall score.
1st. 2nd. 3rd. 4th. 5th. 6th. 7th.
Lopez. 12. 12. 12. 11. 11. 11. 11.
Ondra. 24. 11. 11. 11. 11. 11. 11.
Coleman. 12. 12. 12. 12. 11. 11. 11.
Nurasaki. 22. 22. 22. 22. 22. 20. 20.
Schubert. 13. 28. 45. 64. 85. 90. 95.
M.Mawem. 24. 24. 24. 24. 24. 24. 22.
Duffy. 36. 36. 22. 22. 22. 22. 22.
B.Mawem. 192. 192. 192. 192. 192. 192. 192
Evaluation: the results are similar to the multiplication method, but now Duffy and M.Mawem also have chances to get a medal, and Coleman also could get gold.
A disadvantage of this method is that there seem to be a lot of similar scores, which would result in either silver and/or bronze medals not being awarded or using qualifying rounds to determine the winner. Additionally, for better or for worse, in this method getting first is extremely good for keeping the score low, as in this case the score is just the sum of the numbers.
A benefit of this method is that if someone was worse at one of these events but really good at the others they would still have a chance at a medal (eg. They got 7, 2, 1 and their score would be 10.)
2. Sum of the squares of their scores
1st. 2nd. 3rd. 4th. 5th. 6th. 7th.
Lopez. 66. 66. 66. 59. 59. 59. 59.
Ondra. 56. 53. 53. 53. 53. 53. 53.
Coleman. 62. 62. 62. 62. 53. 53. 53.
Nurasaki. 49. 49. 49. 49. 49. 38. 38.
Schubert. 75. 78. 83. 90. 99. 110. 123.
M.Mawem. 62. 62. 62. 62. 62. 62. 49.
Duffy. 50. 50. 45. 45. 45. 45. 45.
B.Mawem. 192. 192. 192. 192. 192. 192. 192.
I wanted to see how this would change if I added the climber's highest score:
1st. 2nd. 3rd. 4th. 5th. 6th. 7th.
Lopez. 73. 73. 73. 66. 66. 66. 66.
Ondra. 62. 59. 59. 59. 59. 59. 59.
Coleman. 68. 68. 68. 68. 59. 59. 59.
Nurasaki. 55. 55. 55. 55. 55. 43. 43.
Schubert. 82. 85. 90. 97. 106. 117. 130.
M.Mawem. 69. 69. 69. 69. 69. 69. 55.
Duffy. 55. 55. 50. 50. 50. 50. 50.
B.Mawem. 200. 200. 200. 200. 200. 200. 200.
EVALUATION:
Adding the highest score did not make much difference, there only significant things I noticed are that Ondra can now get silver and that Duffy will be awarded gold if Schubert gets first or second. All the changes only happen if Schubert gets first or second.
This scoring system heavily weighs worse scores. A two and a three is about equivalent to an one and a four. A one and a seven is about equivalent to a four and a six or exactly equal to two fives.
Evaluation of the actual method used at the 2020 Olympics:
Benefits:
Do not have to be good at everything; a 1 in any of the climbs means the overall score is the product of the other two.
Works well with the fact none of the climbers are really really good at all three, and that they focus on different climbing skills. -> also helpful for the speed climbers.
Disadvantages:
It is nearly impossible to get gold without getting first in at least one climb (at least shown with by scores from 2020).
2 and 3 is valued the same as 1 and 6
To account for the speed climbers, the judges would have wanted to have a scoring system where getting one 'bad' score would not limit a climber from getting a medal, as long as they did well in the other two. For example not weighting the lower scores so much that it would be impossible to get a medal if a climber got 7 or 8 in one of the climbs.
This is a really interesting point about 2 and 3 being valued the same as 1 and 6 in this scoring system. Does it feel like scoring 2 and 3 is equivalent to scoring 1 and 6? If not, which one feels like the better score?
It certainly looks like coming first in one event means you're more likely to get a medal overall, using the scoring system from the 2020 Olympics. And as you say, Jex, getting one 'bad' score doesn't have too big an impact on a climber who does well in the other two climbing events.
It was very interesting seeing what you all thought about this scoring method. It looks like there are definitely both advantages and disadvantages to it. I don't think any of the solutions mentioned the disadvantage I noticed as I was watching the event, which is that it was very difficult for spectators to work out what was going on!
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.