Number Play | A

Question 6

The first player says 1, 2 or 3. Then the two players take turns adding 1, 2, or 3 to the previous number said. The first player to reach 21 wins!

Play this game several times with your classmate. Are you starting to see the winning strategy? Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say?

Solution

Understand the Question

The game starts at $0$ , and players take turns adding $1$ , $2$ , or $3$ . The first player to say $21$ wins.
By working backward from $21$ , whenever a player reaches a number, the opponent can add at most $3$ and at least $1$ . Thus, in one round of two turns, the total increase can always be controlled to be $1 + 3 = 4$ .
Subtracting $4$ repeatedly from $21$ reveals the key winning numbers that guarantee a win.

Step 1 · Work Backward to Find Winning Numbers

To guarantee reaching $21$ , a player must say $17$ .

If a player says $17$ , the opponent can only reach: $17 + 1 = 18$ $17 + 2 = 19$ $17 + 3 = 20$

From $18, 19,$ or $20$ , the player can add $3, 2,$ or $1$ respectively to reach $21$ .

Similarly, to force the opponent to reach $14, 15,$ or $16$ , the player must say $13$ : $13 + 1 = 14$ $13 + 2 = 15$ $13 + 3 = 16$

From any of these, the player can add the complementary number to land on $17$ .

Step 2 · Identify the Pattern of Winning Numbers

Continuing to subtract $4$ gives all the key target numbers:

\begin{aligned} 21 - 4 &= 17 \\ 17 - 4 &= 13 \\ 13 - 4 &= 9 \\ 9 - 4 &= 5 \\ 5 - 4 &= 1 \end{aligned}

The sequence of winning numbers is: $1, 5, 9, 13, 17, 21$

Step 3 · Determine the Winning Strategy and Player

Since the smallest winning number is $1$ , the first player can always guarantee a win by:

Saying $1$ on the first turn.
In every subsequent round, if the opponent adds $x$ (where $x \in \{1, 2, 3\}$ ), adding $4 - x$ to land on the next winning number ( $5, 9, 13, 17, 21$ ).

Answer

The first player can always win if they play correctly.

The pattern of numbers the winning player should say is $1, 5, 9, 13, 17, 21$ (starting at $1$ and adding $4$ each round).

Common Mistakes

Starting with $2$ or $3$ : If the first player begins with $2$ or $3$ , the second player can seize the winning track by saying $5$ .
Not Complementing to $4$ : If the opponent adds $x$ , the winning player must add $(4 - x)$ to ensure they stay on the target numbers ( $1, 5, 9, 13, 17, 21$ ).

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