To solve this problem systematically, we need to organize the information provided. Let's break it down step by step and create a table that tracks which card each player (P, Q, R, S) picked in each of the four rounds, as well as the points they earned.

Step 1: Setup the Table

We are given the following:

• 4 players: P, Q, R, S
• 5 different colored cards: Red (1 point), Green (2 points), Blue (3 points), Yellow (4 points), White (5 points)
• 4 rounds where each player picks one card per round
• No two players pick the same card in rounds I, III, and IV (condition iv)

Let’s organize this in a table where each row represents a round and each column represents the player’s picks:

Round	P (Points)	Q (Points)	R (Points)	S (Points)
Round I	?	?	?	?
Round II	?	?	?	?
Round III	White (5)	?	?	?
Round IV	?	?	?	?

Step 2: Add Key Information

Based on the information given, we can now fill in certain parts of the table:

• P picked the White card in round III, and scored 5 points (condition iii).
• In round III, no two players picked the same card (condition iv), so Q, R, and S must have picked different cards in that round.
• The total points scored by P and Q across all four rounds are equal (condition v).
• The sum of points scored by P in round I and S in round IV is 5 (condition vi).

Step 3: Logical Deductions

1. Condition ii: No player picks the same card twice across rounds, and P, Q, and R have a maximum score of 12 points across all rounds.

   • This means the highest total score any player can achieve is 12 points. P, Q, and R must therefore score fewer than 12 points each.

2. Condition vi: Green was picked more times than White.

   • Green (worth 2 points) must appear more often in the picks than White (worth 5 points).

3. Condition vii: One card contributes a total of 20 points in the game.

   • This implies that this card must have been picked 5 times (because the total points are $5 \times 4 = 20$).

Step 4: Assign Cards to Players

• Let's hypothesize that the card that contributed 20 points is Green (since it has to be picked more times than White and this would fit the point total).
  • If Green is worth 2 points and was picked 10 times, that contributes exactly 20 points.

**Step 1: Basic Outline and Terms

Consider