Question 6

We can find the winning numbers by working backward from the target number.

Step 1 — The Goal

The game ends when someone says 21. The player who says 21 wins. We can add 1, 2, or 3 to the last number. Let us find numbers that help us win.

Step 2 — Working Backwards

Let us think about the number 21. To say 21, we need a number before it. This number must be 18, 19, or 20. If our friend says 18, we add 3. Then we get 21. If our friend says 19, we add 2. Then we get 21. If our friend says 20, we add 1. Then we get 21. If our friend says 18, 19, or 20, we win. These are bad numbers for our friend. We call them "losing numbers". The player who says them will lose.

Step 3 — Finding Winning Numbers

Now, let us find numbers we want to say. We want our friend to say a "losing number". The "losing numbers" are 18, 19, 20. What number makes our friend say these? Let us try the number 17. If we say 17, our friend adds 1. The number becomes 18. $17 + 1 = 18$ If our friend adds 2, it becomes 19. $17 + 2 = 19$ If our friend adds 3, it becomes 20. $17 + 3 = 20$ All these are "losing numbers" for our friend. So, 17 is a "winning number" for us. We want to say 17.

Let us find the next winning number. We want our friend to say 14, 15, or 16. These would be losing numbers for our friend. What number makes our friend say these? Let us try the number 13. If we say 13, our friend adds 1. The number becomes 14. $13 + 1 = 14$ If our friend adds 2, it becomes 15. $13 + 2 = 15$ If our friend adds 3, it becomes 16. $13 + 3 = 16$ These are all losing numbers for our friend. So, 13 is a "winning number" for us.

We see a pattern here. The winning numbers are 21, 17, 13. Each winning number is 4 less than the last. Let us list all winning numbers. We subtract 4 each time. $21 - 4 = 17$ $17 - 4 = 13$ $13 - 4 = 9$ $9 - 4 = 5$ $5 - 4 = 1$

$\boxed{\text{Winning numbers: } 1, 5, 9, 13, 17, 21}$

Step 4 — The Winning Strategy

The winning player must always say these numbers. The first player says 1, 2, or 3. If the first player says 1, they start well. If they say 2 or 3, they do not. So, the first player should start with 1. Then, the current number is 1. Our friend will add 1, 2, or 3. The number will become 2, 3, or 4. We want to reach the next winning number, 5. If our friend says 2, we add 3. This makes 5. If our friend says 3, we add 2. This makes 5. If our friend says 4, we add 1. This makes 5. Our move and friend's move add to 4. This lets us land on the next winning number. We keep doing this until we say 21.

Answer

(i) The first player can always win if they play correctly. (ii) The winning player should always say numbers from this pattern: 1, 5, 9, 13, 17, 21.