Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?

Question

Homi · Accepted Answer

Understand the Question

In a game where players take turns adding a number from $1$ to $3$ to reach a target of $22$ , working backwards helps identify target (safe) positions.
Since the sum of minimum and maximum possible additions per round is $1 + 3 = 4$ , controlling the total in steps of $4$ guarantees a win.
Working backwards from the target $22$ by repeatedly subtracting $4$ gives the winning sequence: $18, 14, 10, 6, 2$ .
The first player can always guarantee a win by landing on these safe numbers on every turn.

Step 1 · Find the Safe Target Numbers

To guarantee reaching $22$ , determine the target numbers by working backward.

If a player leaves the total at $18$ , the other player can only reach $19, 20$ , or $21$ , allowing the first player to immediately reach $22$ .

Since the range of numbers is $1$ to $3$ , the complementary sum per round is $1 + 3 = 4$ . Subtracting $4$ sequentially from $18$ gives all previous target positions:

$18 - 4 = 14$

\begin{aligned} 14 - 4 &= 10 \\[0.6em] 10 - 4 &= 6 \\[0.6em] 6 - 4 &= 2 \end{aligned}

Thus, the safe target numbers are $2, 6, 10, 14,$ and $18$ . Diagram 1

Step 2 · Execute the Winning Strategy

First Move: The first player starts at $0$ and adds $2$ to land on the first safe number: $0 + 2 = 2$
Subsequent Moves: Whenever the second player adds $x$ (where $x \in \{1, 2, 3\}$ ), the first player responds by adding $(4 - x)$ :

\begin{aligned} 1 + 3 &= 4 \\[0.6em] 2 + 2 &= 4 \\[0.6em] 3 + 1 &= 4 \end{aligned}

This ensures the total increases by $4$ each round, landing on $6, 10, 14, 18,$ and finally $22$ to win the game.

Answer

The first player has a winning strategy:

Start by adding $2$ on the first turn to reach $2$ .
On every subsequent turn, if the second player adds $x$ , the first player adds $4 - x$ so that their combined sum is always $4$ .
This forces the total to reach $2, 6, 10, 14, 18,$ and finally $22$ , guaranteeing a win.

Common Mistakes

Wrong Starting Move: Choosing $1$ or $3$ at the start instead of $2$ hands the advantage to the second player, who can then control the multiples of $4$ plus $2$ .
Incorrect Complement: Adding a fixed number regardless of what the opponent plays instead of ensuring the two-turn sum equals $1 + 3 = 4$ .

Question 33

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