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Probability and Logical Reasoning: Card Picking Ga...
Math Problem Statement
Directions for questions 6 to 10:


Answer the questions on the basis of the information given below.


Four persons P, Q, R and S are playing a game of blindly picking the cards from a box. This box contains five cards, each of a different colour among Red, Green, Blue, Yellow and White, and these five cards fetch 1, 2, 3, 4 and 5 points respectively. The game is being played in four rounds. In each round, each person randomly takes out one card from the box and places it back into the box. Based on the colour of the card, they have been given points as mentioned above.


Further it is known that,


i. Each card was picked by at least one person and each card was picked a different number


ii. None of P, Q and R picked the same card in any two rounds and the maximum points


scored by any of P, Q and R in all the four rounds put together are 12 iii. Among the four rounds, the maximum points were scored in round Ill and P picked the white


card in round Ill.


iv. In each of rounds I, Ill and IV no two players picked the same card.


v. The points scored by P and that by Q in all the four rounds put together are equal. vi. The green card was picked more times than the white card. The sum of the points scored by


P in round I and that scored by S in round IV is 5.


vii. One of the five cards contributed a score of 20 to the total points scored in all the four


rounds put together. Make a table

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

Four persons P, Q, R, and S are playing a game of blindly picking cards from a box. This box contains five cards, each of a different colour among Red, Green, Blue, Yellow, and White, and these five cards fetch 1, 2, 3, 4, and 5 points respectively. The game is being played in four rounds. In each round, each person randomly takes out one card from the box and places it back into the box. Based on the colour of the card, they have been given points as mentioned. Further, it is known that each card was picked by at least one person and each card was picked a different number of times. None of P, Q, and R picked the same card in any two rounds, and the maximum points scored by any of P, Q, and R in all four rounds put together are 12. Among the four rounds, the maximum points were scored in round III, and P picked the white card in round III. In each of rounds I, III, and IV, no two players picked the same card. The points scored by P and that by Q in all four rounds put together are equal. The green card was picked more times than the white card. The sum of the points scored by P in round I and S in round IV is 5. One of the five cards contributed a score of 20 to the total points scored in all four rounds put together.

Explore a card-picking game involving four players and five differently colored cards. Solve a logical reasoning problem by analyzing rounds, points, and conditions to determine which card each player picked. Suitable for high school students, covering probability, combinatorics, and logical deduction.

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