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

The game starts at $0$ and two players take turns adding $1$ , $2$ , or $3$ . The first person to reach $22$ wins.
By working backwards from $22$ , if you leave your opponent at $18$ , whatever they add ( $1, 2,$ or $3$ ) brings the total to $19, 20,$ or $21$ , allowing you to reach $22$ on your next turn.
Since the minimum addition is $1$ and maximum is $3$ , you can always pair your turn with the opponent's turn to sum to $1 + 3 = 4$ .
Subtracting $4$ repeatedly gives the safe winning positions: $2, 6, 10, 14, 18, 22$ .

Step 1 · Find the Winning Positions (Safe Numbers)

Working backwards from the target $22$ :

From $19, 20,$ or $21$ , adding $3, 2,$ or $1$ immediately wins.
Leaving the opponent at $18$ forces them to reach $19, 20,$ or $21$ , guaranteeing your win.

Since players can add between $1$ and $3$ , the sum of a two-turn round can always be controlled to $1 + 3 = 4$ . Subtracting $4$ successively from $18$ gives the earlier safe positions:

$18 - 4 = 14$

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

Thus, the safe positions are $2, 6, 10, 14, 18$ . Diagram 1

Step 2 · Formulate the Winning Strategy

1. First Turn: The first player must add $2$ to land on the first safe number:

$0 + 2 = 2$

2. Subsequent Turns: Whenever the second player adds $x$ , the first player responds by adding $(4 - x)$ so that each round increases the total by $4$ :

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

This ensures the first player always reaches $6, 10, 14, 18,$ and finally $22$ to win.

Answer

The first player has the winning strategy:

Add $2$ on the first turn.
On every subsequent turn, if the second player adds $x$ , add $4 - x$ to maintain the winning chain ( $2 \to 6 \to 10 \to 14 \to 18 \to 22$ ).

Common Mistakes

Wrong Starting Move: Adding $1$ or $3$ on the first move instead of $2$ allows the second player to take control of the safe positions.
Fixed Addition Mistake: Adding a fixed number each turn instead of adapting with $(4 - x)$ based on what the other player added.

Question 33

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