Ms. McCarthy from St. Francis Xavier in the United States Ӭsent in the following contribution:

We determined that the gold medal totals were used to determine what place each of the 10 countries listed came in. If there was a tie, they went to the silver medal count to choose which team placed higher.

Another way to order the list would be to go with the total medal count. While it wouldn't change the top few, it would certainly change some others, especially Australia.

We think that no matter how you manipulate the results, USA would always come out on top because they won the most medals in every category.

We had a number of very good ideas coming from Roseville College in Australia.

Firstly from Mia:

1. Is your own nation in the list? Yes my nation is Australia.
2. How do you think the positions have been decided? The positions have been decided by the number of gold medals won by each country.
3. Could the results be presented differently to give another nation the top place? The results could be presented in order of the total number of medals won, but America would still be in first place. 
4. How would this affect other results in the table? Some countries will move down the table and some countries will move up the table like Australia. Australia would move to 6th position.

Ella wrote:

The way that the Olympics place countries is by the amount of gold medals your country wins. Another way you can order the top 10 countries is by alphabetical order. (I never said it would be fair). Another way is the total amount of medals your country wins.
The way it would affect the scores on the table is that the countries that have a first letter in the alphabet that is far down in the alphabet would be low in the ranking. It would take the fun of the Olympics for those countries because they have no chance of winning.
Then my second reason is that we should place the countries by how many medals they win altogether not how many gold medals they win. This is because a country may be able to easily get heaps of bronze and silver medals but still lose because they have very few gold medals. If they order countries according to how many medals they have altogether it would give other countries a chance of winning.

Imogen had these thoughts:

The positions have been decided based on how many Gold medals the country won. If the ranking was based on how many Bronze medals were won, Russian Fed. would be in the top place and every other team would be in a different position.  Australia and France would both share 6th place.

Hannah also answered the questions:

The country that has the most amount of medals in total could be at the top.
The country that tried the hardest could be at the top.
The country that went in the most events could be at the top.
How would this affect other results in the table? 
Countries who might not have got into the top 10 might get into the one above.

Alice wrote the following:

1. Yes, Australia is on the table.
2. By the number of gold medals.
3. Yes, If the countries where ranked by total number of medals but the USA would still be on top. 
4. Ranks would swap around and maybe even new countries would appear on the table.

Ashley showed how the resuts might be changed:

Australia came tenth on the list with 7 gold medals, 16 silver medals and 12 bronze medals. I think the positions have been decided by the number of gold medals the country has: for example, the United States has 46 gold medals and China has 38. The United States came 1st and China came 2nd. If there was a tie in gold medals they would look on how many silver ones they have, like they have done for Italy and Hungary. You can change it by looking at the total medal tally which would change the positions on the leaderboard.

Leaderboard with total instead of gold medals:

United States, 104
China, 88
Russia, 85
Great Britain, 65
Germany, 44
Australia, 35
France, 34
South Korea, 28
Italy, 28
Hungary, 17

South Korea came before Italy because they had more gold medals as well as
totals. 

Thank you, Roseville students for those excellent solutions.

Abhishek from Singapore International School, Mumbai in IndiaӬ wrote:

I believe that the positions have been decided by the number of gold medals first, then silver, then bronze (in descending order).
I believe gold is first because Great Britian is ahead of Russia even though Russia is ahead on silver and bronze.
Silver is second because Germany is ahead of France.

Could the results be presented differently to give another nation the top place?
I tried calculating the median of the gold, silver and bronze for each country, but the 1st place remains the same, that is US which is 29 (29, 29, 46) because the median comes ahead of China (23, 27, 38).
If you rank by the highest number of bronze medals, Russia comes first. South Korea comes down dramatically on most bronze medals. Australia comes up by three places.

Thank you for all the contributions.  We hope to see you sending in solutions to other NRICH activities.