Prompt Title: 3-match Game Possibilities.

To determine all the possibilities of you winning the three match game with each match rewarding points from 0 to 3 for 4th to 1st place, we need to consider all the possible outcomes of the three matches and calculate the total points you would earn in each scenario.

Assuming that you are competing against other players, there are a total of 4^3 = 64 possible outcomes of the three matches, as each match can result in one of four scores (0, 1, 2, or 3).

We can use a table to organize all the possible outcomes and calculate the total points you would earn in each scenario. Let's denote your total points in the three matches by P.

To determine the cases in which you would win the three match game with four players and each match rewarding points from 0 to 3 for 4th to 1st place, we need to look at the table of all possible outcomes and calculate your total points (P) in each scenario.

Assuming that all players have an equal chance of scoring in each match, there are 4^3 = 64 possible outcomes of the three matches, as each match can result in one of four scores (0, 1, 2, or 3).

To win the game, your total points (P) must be greater than the total points of all other players.

Let's consider some examples:

• If all other players score 0 points in all three matches, you would win the game if your total points are greater than 0.
• If all other players score 1 point in all three matches, you would win the game if your total points are greater than 3.
• If all other players score 2 points in all three matches, you would win the game if your total points are greater than 6.
• If all other players score 3 points in all three matches, you would win the game if your total points are greater than 9.

These are just a few examples, but you can use the table to calculate your total points in each scenario and determine whether you would win the game or not. Remember, you need to have the highest total points among all players to win the game.

If you get 1st place in two matches, then your minimum possible score is 6 points, even if you get last place in the third match. This is because each match awards at least 1 point for 1st place, and if you get 1st place in two matches, you will have earned at least 2 points from those matches.

To determine the cases where you would definitely win the game with 6 points, even in the worst-case scenario, we need to consider all the possible outcomes of the third match where you don't get 1st place.

Assuming that all other players have an equal chance of scoring in each match, there are 3 possible outcomes of the third match where you don't get 1st place:

1. You get 2nd place in the third match and earn 2 points.
2. You get 3rd place in the third match and earn 1 point.
3. You get 4th place in the third match and earn 0 points.

Let's look at each case:

1. If you get 2nd place in the third match and earn 2 points, your total score would be 8 points (6 + 2). This would be the highest possible score and you would definitely win the game.
2. If you get 3rd place in the third match and earn 1 point, your total score would be 7 points (6 + 1). In this case, you would win the game if all other players get a total of 6 or fewer points in the three matches. If any other player gets more than 6 points, they would win the game instead of you.
3. If you get 4th place in the third match and earn 0 points, your total score would be 6 points (6 + 0). In this case, you would win the game if all other players get a total of 5 or fewer points in the three matches. If any other player gets more than 5 points, they would win the game instead of you.

So, if you get 1st place in two matches, you would definitely win the game with 6 points if either of the following conditions is met:

• You get 2nd place in the third match and earn 2 points.
• You get 3rd or 4th place in the third match and all other players get a total of 6 or fewer points in the three matches (or 5 or fewer points if you get 4th place in the third match).