I02-17

Question

Four friends, Anna, Bob, Clara, and Dan, trade three colors of marbles according to agreed rules. They have decided that any marble of one color can be exchanged for another of the same color. Additionally, marbles follow a fixed value system, and trades can occur according to the given ratios or any values derived from them: two red marbles have the same value as one blue marble, and two blue marbles have the same value as one green marble.
 
There are 13 marbles in total, divided as follows: Anna has one red and two green marbles; Bob has one red and one blue; Clara has two blue marbles and one green; Dan has four red marbles and one blue.
 
Select the maximum and minimum number of marbles that any of the friends can have after exactly three trades are made. A trade involves two people exchanging any number of marbles according to the rules. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

We have the following distribution of marbles:
 
         Anna     Bob     Clara     Dan        Red   1   1   0   4       Blue   0   1   2   1       Green   2   0   1   0      
 
We are given that (Green) = 2(Blue) = 4(Red). Since a green marble has the highest "value," Anna, with two green marbles, has the highest total value of marbles, equivalent to 9 red marbles. However, since there are not 9 red marbles available across all players, she cannot achieve that count. Instead, Anna could trade 1 green marble with Dan for 4 red marbles and 1 remaining green marble with Clara for 2 blue marbles.
 
This would give Anna 6 additional marbles for her 2 greens, increasing her total to 7 marbles. For the third trade, Bob and Dan could simply exchange their blue marble between them, as it does not affect Anna’s total count. Therefore, the maximum number of marbles Anna can have after the trades is 7.
 
As for the minimum number of marbles, note that having 0 marbles is not possible because you cannot completely get rid of marbles—you are always trading some marbles for others, so the total cannot reduce to 0 (assuming you didn’t start with 0). However, it is possible to end up with just 1 marble. For example, if you have 2 red marbles, you can trade them for 1 blue. If you have 4 red marbles, you can trade them for 1 green. Similarly, if you have 2 blue marbles, you can trade them for 1 green. Likewise, a combination of 2 red and 1 blue marbles can be traded for 1 green. Thus, under specific conditions, it is possible to reduce the total to 1 marble.
 
However, notice that none of these cases apply to the current situation, so ending with just 1 marble is also not possible.
 
Therefore, the minimum must be 2 marbles. For example, Bob already has 2 marbles, and if he doesn’t participate in the trades at all while the trades occur only among the other three people, he would still have 2 marbles.

Correct answer:

Maximum  "7"

Minimum  "2"

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity.

Unknown5612 · Answer

They have decided that any marble of one color can be exchanged for another of the same color - this is confusing - I thought, for e.g., green could only be exchanged for another green (which doesn't change anything - hence got confused during the test).

Bunuel · Answer

Yes, the question says same-color marbles can be exchanged (e.g., red for red), and after that, it introduces additional trading options across colors based on value: "   Additionally  , marbles follow a fixed value system, and   trades can occur   according to the given ratios or any values derived from them: two red marbles have the same value as one blue marble, and two blue marbles have the same value as one green marble.  "

I02-17

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