Climbing conundrum
Five children are taking part in a climbing competition with three parts, where their score for each part will be multiplied together. Can you see how the leaderboard will change depending on what happens in the final climb of the competition?
Problem
There are three climbing walls in the playground at Winterbrook School. Five children are having a competition to see how far they can climb up each one.
They have decided that whoever climbs the furthest on each wall should get a score of 1, then the person who comes second should get a score of 2, and so on. The three scores for each child will then be multiplied together, and the person with the lowest score will win.
Here are their scores for the first two climbing walls:
| Wall 1 | Wall 2 | Multiplied scores | |
|---|---|---|---|
| Anu | 3 | 4 | |
| Bryn | 1 | 5 | |
| Clare | 2 | 3 | |
| Duncan | 4 | 1 | |
| Elise | 5 | 2 |
At the moment, who is in the lead?
If the scores were added together, rather than multiplied, would this change the order? How?
On the third wall, after the first four children have climbed, the scores look like this:
| Wall 1 | Wall 2 | Wall 3 | Multiplied scores | |
|---|---|---|---|---|
| Anu | 3 | 4 | 1 | |
| Bryn | 1 | 5 | 3 | |
| Clare | 2 | 3 | 2 | |
| Duncan | 4 | 1 | 4 | |
| Elise | 5 | 2 |
Have a go at calculating the multiplied scores at this point in the competition. (Remember, these scores will only be accurate if Elise comes fifth on Wall 3, because if she overtakes anybody else then their score for Wall 3 will change.)
Elise starts her climb. Anu says to the others, "I’m joint first out of the four of us, so if Elise overtakes me then I’ll still finish in the top three."
Do you think that Anu is probably right? Can you explain your thinking?
Have a go at working out what everybody's scores will be if Elise climbs the highest on Wall 3. Compare this to what would happen if Elise came fifth on Wall 3. (You might also like to work out what would happen if Elise came second, third or fourth, to compare all the possible outcomes.)
How would the outcome have been different if the scoring is reversed (5 for the best, 1 for the worst)?
How would the outcome have been different if the scores were added rather than multiplied?
Which scoring system do you think is fairest? Why?
Student Solutions
Thank you to everybody who sent in their thoughts about this problem. We received a lot of excellent solutions, so unfortunately we don't have room to feature them all on this page.
Lucas, Oliver and Derek from Kohia Terrace School in New Zealand answered our questions about the scores for the first two climbing walls:
At the moment, who is in the lead?
Anu=3×4=12
Bryn=1×5=5
Clare=2×3=6
Duncan=4×1=4
Elsie=5×2=10
Duncan is in the lead.
If the scores were added together, rather than multiplied, would this change the order? How?
Yes it would change the order because Clare and Duncan would tie
I wonder which scoring system seems fairer at this point? Does it feel like Duncan's 1st place and 4th place scores should beat Clare's 2nd and 3rd place scores, or does it feel like they should tie with those scores?
Joey from Eden Primary in the UK worked out everybody's ranking at the end of Wall 2:
If you multiply the wall 1 and wall 2 scores, you get Anu with 12, Bryn with 5, Clare with 6, Duncan with 4 and Elise with 10. So, Duncan is 1st, Bryn is 2nd, Clare is 3rd, Elise is 4th and Anu is 5th.
If you add the scores then Anu has 7, Bryn has 6, Clare has 5, Duncan has 5 and Elise has 7. So, Clare and Duncan are joint 1st, Bryn is 3rd and Anu and Elise are 5th.
Joey then thought about what might happen after Wall 3, depending on where Elise ends up:
Anu is wrong. If Elise comes 1st then he would come second and his total score would be 24 (3×4×2). Bryn would get 20 (4×5×1). Clare would get 18 (2×3×3). Duncan would get 20 (4×5×1). Elise would get 10 (5×2×1). Elise 1st, Clare 2nd, Duncan and Bryn 3rd and Anu 5th.
If Elise comes 5th on wall 3, she would have a score of 50, Anu 12, Bryn 15, Clare 12 and Duncan 16. So, Anu and Clare 1st, Bryn 3rd, Duncan 4th and Elise 5th.
If Elise comes 2nd on wall 3 then Anu = 12, Bryn = 20, Clare = 18, Duncan = 20 and Elise = 20. So, Anu 1st, Clare 2nd and Duncan, Elise and Bryn 3rd.
If Elise comes 3rd then Anu = 12, Bryn = 20, Clare = 12, Duncan = 20 and Elise = 40. So, Anu and Clare 1st , Bryn and Duncan 3rd and Elise 5th.
If Elise comes 4th then Anu = 12, Bryn = 15, Clare = 12, Duncan = 20 and Elise = 40. So, Anu and Clare 1st, Bryn 3rd, Duncan 4th and Elise 5th.
Jasper and Rory from Banstead Prep School in England explained why Anu is wrong:
We think that Anu is wrong because her score for the first 2 walls are 12 which is the highest score. This means that if she gets a higher number her score will be larger.
It certainly seems like going from 1st place to 2nd place on the final wall would make a big difference to Anu's overall score.
Jaelyn from Brentwood Prep School in the UK looked at the leaderboard after the first two walls, and noticed something about what happens when we multiply the scores compared to adding them:
The person in the lead for scores 1 and 2 is: Duncan. The scores would change if they were added because if they were added, they usually have a lower score. For example, Anu’s score when multiplied is 12, because he has 3 for his 1st climb and 4 for the second. But when added, Anu’s score is 7. So the scores change, but Duncan is still in the lead.
This is a good point, Jaelyn - multiplying two numbers usually gives a bigger result than adding those two numbers. I wonder if this is always the case?
Joey also thought about what difference it would make if the scores were added:
The difference is that if someone got 3 and 3 and someone got 5 and 1 they would draw. I think this is fairer because otherwise a 1 is too good and a 5 is too bad.
The multiplication scoring system certainly does seem to reward scores of 1, and penalise very high scores.
Ryan from Kohia Terrace School agreed with this:
I think that adding the scores with normal scoring would be the fairest because for example if one person got 2nd and 2nd and another person got 3rd and 1st, when you multiply them then the 2nd person would win but if you add them, then it would be a tie. I think that the correct outcome would be the tie because if person 2 gets good and bad scores and person 1 gets consistent average scores, I think that they are equal. This is why adding the scores would be better than multiplying the scores (in my opinion).
Niam from Banstead Prep had a different idea about how the scoring could be done:
The scoring system should be the mean (average) of the scores.
This is a good idea, but I think this would end up giving the same ordering as the adding scoring system - can you see why?
Isaac from Kohia Terrace School preferred the multiplication scoring system, and explained why:
I think the fairest scoring system would be multiplying with normal scoring. For example if someone got 3 and 2, then someone got 1 and 4, if you multiply it it would be 6 and 4. if you add it it would be a tie (5 and 5). Then in my opinion there should be a winner so that's why multiplying would be the fairest scoring system.
Jasper and Rory from Banstead Prep looked at how the scores would change if we reversed the scoring system:
The current scores are (before Elise climbs):
| Name | Wall 1 | Wall 2 | Wall 3 | Multiplied scores |
| Anu | 3 | 4 | 1 | 12 |
| Bryn | 1 | 5 | 3 | 15 |
| Clare | 2 | 3 | 2 | 12 |
| Duncan | 4 | 1 | 4 | 16 |
| Elise | 5 | 2 | ? |
If the score were reversed it would look like this:
| Name | Wall 1 | Wall 2 | Wall 3 | Total |
| Anu | 3 | 2 | 5 | 30 |
| Bryn | 5 | 1 | 3 | 15 |
| Clare | 4 | 3 | 4 | 48 |
| Duncan | 2 | 5 | 2 | 20 |
| Elise | 1 | 4 | ? |
Clare would win. Because the scores were reversed the winner would have the highest score.
Golden Eagles Class from Anston Greenlands Primary School in the UK also looked at the reverse multiplication scoring system, and they worked out what might happen after Elise's climb. They then explained which scoring system seemed fairest to them:
We found out that if the highest climber is given a score of 1 and the three scores are multiplied, then either Elise would win, Anu would win or Anu and Clare would tie.
If the scoring is reversed and the scores are multiplied then Clare would win, no matter how high Elise climbed.
If the scores are added, Clare would win, or Anu and Clare would tie or Clare and Elise would tie.
Ella and Joe
Joe said – I feel like Anu or Clare should win because they both win eight times out of the possible fifteen results. Adding their scores seems like the fairest way.
Ella said – I think that having a score of 1 for the person who goes highest is fairest. I think it is also fair to multiply the score. That way, Anu and Clare have a good chance of tying for first place.
Emilia and Imogen
We think that adding is the fairest system because it makes a lot more sense. Clare won every time when the scores were reversed and multiplied but we think Anu deserves to tie with Claire because he is first or second on the third wall.
Matilda and Bella
We think it makes sense to reverse the scores then multiply. Your aim is to get the highest number of points. With this system, Claire would be the winner, no matter how well Elise did on the final wall. This means she deserves that win by miles (fair and square).
It is interesting to hear all your different ideas about this. Deciding on the fairest scoring system certainly isn't a simple task!
Lots of children thought hard about the benefits and drawbacks of the multiplication scoring system. Daniel from Brentwood Prep School said:
As for the fairest scoring system, the multiplied scores provide a greater penalty for poor performances, which can lead to a more dramatic outcome. The added scores provide a more balanced approach, where each wall has an equal weight. The reversed scoring system (5 for the best, 1 for the worst) would also produce a different outcome.
Ultimately, the fairest scoring system depends on the goals and preferences of the competition organizers. If they want to emphasize consistency across all walls, the added scores might be a better choice. If they want to reward exceptional performances and penalize poor ones, the multiplied scores could be more suitable.
This is a very good point, Daniel. This scoring system is actually from the 2020 Olympics, where all three 'walls' were very different. In that case, I think the organisers were trying to reward exceptional performance in any one of the three events.
Joshua, Yuichiro, Tom and Daichi from St. Mary's International School in Tokyo, Japan noticed that coming first on a wall gave a climber a big advantage overall:
We think that adding would be fairer than multiplying the scores because people could do very well on 1 wall and do bad in the other ones, as in multiplying, if you get 1st place, you start with a huge advantage on the second wall. In adding, however, even if you score 1st, you must score low on other walls too to win.
Take a look at Joshua, Yuichiro, Tom and Daichi's full solution to see the tables of results that they calculated.
William from Kohia Terrace School explained that going from first place to second place on a wall would double a climber's score:
If the scores were added together, rather than multiplied, would this change the order? How?
Yes it would because instead of multiplying, which increases it a lot, it just adds.
Do you think that Anu is probably right? Can you explain your thinking?
No because by dropping down to second it doubles Anu's score which makes him go down very far.
Which scoring system do you think is fairest? Why?
Adding because if you get second instead of first your score won't be doubled and you'll statistically speaking have a better chance.
Sithuki from Pristine Private School in Dubai, UAE generalised this idea to different multipliers:
Which scoring system do you think is fairest? Why?
The Adding System.
- While calculating scores, multiplication has much greater impact than addition as it amplifies differences significantly
- Even small variation in multipliers can drastically shift the rankings
It's easy to see that going from 1st to 2nd place on a wall will double your overall score. But I wonder what happens if you go from 2nd to 3rd place on a wall? Or 3rd to 4th? Which step down has the biggest impact on your overall score?
Well done to the following students who also sent in similar solutions to this problem: Rohan, Elspeth, Darcie, Arushi, Cooper, Zain, Arthur, Logan, Naomi, Kosara, Savannya and Grace from Banstead Prep; Daniel and Finn from Brentwood School; Taym from Horfield Primary School in Bristol, UK; Temi from St Philip Westbrook Church of England Aided Primary School in England; Charlie and Logan from Kilmore CNS in Ireland; Jace from Pristine Private School in Dubai, UAE; Hazelnut and Rome at Clifton Hill Primary School in Australia; and Noah and Sidney from Eden Primary N10 in England. It was really interesting to see what you all thought about this scoring system.
Teachers' Resources
Why do this problem?
This problem provides an engaging context for exploring multiplication, and particularly for noticing that the product of two numbers can get bigger very quickly as the numbers increase. The fact that Elise's score on the final wall can drastically change the positions of the other climbers 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.
Possible approach
As a whole class, talk through the first table and make sure that all children understand how the scoring system works. The idea of multiplying the scores for each wall to get the final scores will need to be explained, but learners might also be confused by the fact that a lower score is better. You might like to give some scoring methods for different sports as a reference for this, such as how the winner of a race will be the athlete with the shortest time.
Give children some time to calculate the multiplied scores after the first two walls have been climbed, and encourage them to compare these to the scores that would be produced by adding the numbers together instead of multiplying them. Learners can then begin to consider the differences between these scoring systems - what sorts of scores are generally better if the numbers are being multiplied? Or if the numbers are being added? As you walk around the room, listen out for children who have noticed that a score of 1 on any wall will be heavily rewarded in the multiplication scoring system.
Bring the whole class back together to discuss what the children have found out, before explaining the next part of the task. For the second table, learners can fill in the multiplied scores as they stand, but they will need to find a way of recording different possible scores based on Elise's position on Wall 3. Some children might draw their own tables for this, whereas others might prefer to make notes in a different way. You could give learners the freedom to choose their own way of recording their findings, or you could give them a sheet of pre-made tables if you'd prefer them to just focus on the multiplication. Every child should be given time to complete at least two tables - one where Elise comes first, and one where she doesn't - in order to give them the 'aha!' moment of seeing just how much the scores of the other children can change depending on Elise's score on Wall 3.
At the end of the activity, bring the whole class back together to discuss what they've noticed. What would their strategy be if they were climbing? Is it better to do equally well on all three walls or better on one and worse on the others? You might then like to share some information about the 2020 Olympic climbing, which is at the bottom of this page.
Key questions
Who is in the lead now? What order are the five climbers in?
What would happen if Elise came first on Wall 3? How would everybody else's scores for Wall 3 change?
If everybody's scores for Wall 3 (apart from Elise's) increased by 1 (because they were overtaken by Elise), would their order be the same on the leaderboard? Why/why not?
If gold, silver and bronze medals were given out, what do you notice about who would get the medals depending on how well Elise climbs? (Interestingly, Duncan doesn't get a medal if Elise comes fifth, but if Elise comes third (overtaking him) then Duncan gets a medal...!)
Is it best to be first on one wall and then not do very well on the other two walls, or to do quite well on each wall? Why?
How would the scores change if we used a different scoring system? What if we added the numbers instead? Or what if...? (Children will have their own ideas about different possible scoring systems, and might like to look at the pros and cons of each.)
Possible extension
A lot of time could be spent on critiquing this scoring system and trying out alternative scoring systems. Children might like to write to the IOC with their suggestions!
Climbing Complexity is a version of this task using the actual scores from the 2020 Olympics, which might be suitable as an extension for some learners.
Possible support
Some children will need support with understanding this task, and will benefit from focusing on creating just two final tables - one where Elise comes first on the final wall and one where she comes last. Multiplication tables (or calculators) can also be provided, as the focus in this task isn't about answering the multiplication questions but is instead about being able to notice patterns such as how multiplying by 1 gives a very small answer, and how multiplying slightly bigger numbers can give a much bigger answer.
More information about this scoring system:
This task uses similar scoring rules to the ones used in sport climbing in the 2020 Olympic Games, where there were three different disciplines – speed climbing, bouldering and lead climbing – but only one set of medals, so an athlete's scores for each of the three disciplines were multiplied together. In the men's final of those Olympics, this scoring system led to some very interesting things happening. If you'd like to investigate this, take a look at the problem Climbing Complexity.
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.