Number Play · Lesson 7 of 8
Games and Winning Strategies
“Work backwards from a target and use complementary moves to keep control of a number game.”
• Apply the rules of a target-number game and distinguish the current total from a move. • Work backwards to identify totals from which a win can be forced. • Explain the winning sequence for the games to 21 and to 99. • Use complementary replies to preserve a winning pattern despite an opponent’s choices. • Adapt the reasoning when the target or the allowed moves change.
A strategy explains what to do after every reply
Play a game with a partner. The starting total is 0, and the first player adds 1, 2, or 3, announcing the new total. The next player adds 1, 2, or 3 to that total, and the turns alternate. The player who reaches exactly 21 wins. The initial statement of 1, 2, or 3 is simply the first move from zero.
The announced number and the amount added are different. If the total is 8 and you add 3, you announce 11. You have not added 11. Because all moves increase the total, passing 21 cannot lead back to a win; keep moves within the target and aim to reach it exactly. Write each move and total during your first few games so that there is no confusion about the rules.
A plan that guarantees a win when followed correctly, no matter which legal moves the other player chooses.
A lucky game does not establish a winning strategy. A useful plan tells you how to reply to each possible move, including an inconvenient one. Start by playing several rounds, then look for totals that seem especially useful. Working backwards from the finish can reveal why those totals matter.
Work backwards from 21
If it is your turn and the current total is 18, 19, or 20, you can win immediately: add 3, 2, or 1 respectively. This makes 17 a valuable total to announce. After you say 17, your opponent must move to one of 18, 19, 20, leaving you a move to 21. You are planning around all three replies, not guessing which one they prefer.
| Total you have announced | Opponent’s move | Opponent’s new total | Your winning reply |
|---|---|---|---|
| 17 | 1 | 18 | Add 3 to reach 21. |
| 17 | 2 | 19 | Add 2 to reach 21. |
| 17 | 3 | 20 | Add 1 to reach 21. |
Problem
You have just announced 17. Explain why the opponent cannot prevent your next turn from reaching 21.
- 1.The opponent can add only 1, 2, or 3, so their new total must be 18, 19, or 20.
- 2.Those totals are respectively 3, 2, or 1 below 21. Every required reply is allowed.
- 3.Thus all possible replies lead to an immediate win for you. This is why announcing 17, rather than merely arriving near 21, gives control of the finish.
Use the same reasoning to ask how to force 17. If you announce 13, your opponent moves to 14, 15, or 16, and you can add 3, 2, or 1 to reach 17. Repeat the argument backwards. The useful totals are 1, 5, 9, 13, 17, 21. Each is four above the preceding one.
Complementary replies preserve the pattern
A pair of moves can total four in three ways: 1 + 3, 2 + 2, or 3 + 1. Your reply complements the opponent’s move to make four. If you have announced one of the landmarks and follow this rule, the next total you announce is the next landmark. This lets you preserve control without knowing the opponent’s move in advance.
| Opponent adds | You add | Total added over the two turns |
|---|---|---|
| 1 | 3 | 4 |
| 2 | 2 | 4 |
| 3 | 1 | 4 |
Problem
The first player opens with 1. The second player then adds 3, 1, 2, 3, and 1 on their turns. Find the first player’s replies.
- 1.After the opening 1, an opponent addition of 3 produces 4. Reply with 1 to announce 5.
- 2.The opponent adds 1 to get 6. Reply with 3 to reach 9. Then the opponent adds 2 to get 11; reply with 2 to reach 13.
- 3.The opponent adds 3 to get 16. Reply with 1 to reach 17. Finally the opponent adds 1 to get 18; reply with 3 to reach 21.
- 4.The first player’s announced totals remain 1, 5, 9, 13, 17, 21. At every stage the reply is legal and the two moves add four.
The first player can force a win by opening with 1. Opening with 2 or 3 does not preserve this guarantee: the other player can take 5 and then use the same four-at-a-time pattern. A player who misses a landmark may regain control if the opponent makes a mistake, but that possibility is not a guaranteed strategy.
Four is the combined increase over two turns. Each individual move must still be 1, 2, or 3. The strategy depends on a legal reply that complements the opponent’s move, not on announcing any desired total without following the rules.
Change the move range and work backwards again
In the second game, start at 0 and add any whole amount from 1 through 10 on each turn. The first player to reach exactly 99 wins. The old gap of four no longer fits the rules. Instead, complement an opponent’s addition to make eleven: 1 pairs with 10, 2 with 9, and so on, up to 10 with 1.
Work back from the target in steps of eleven: 99, 88, 77, 66, 55, 44, 33, 22, 11, 0. Here zero is already a landmark before the first turn. The second player can always bring the total to 11 after the first move, because a first move between 1 and 10 leaves an allowed difference to 11. The second player can then maintain every later landmark.
Problem
The first player opens with 7. Later, after the second player has reached 88, the first player adds 8. What should the second player do?
- 1.For the first pair of turns, 11 − 7 = 4. The second player adds 4 and announces 11.
- 2.At the later landmark 88, the first player adds 8, announcing 96.
- 3.The second player adds 11 − 8 = 3, which is allowed, and reaches 99. The same complementary reply works at both stages.
Notice that the winning player changed. In the 21 game, the first player could claim the first useful total. In the 99 game, the landmark spacing makes the second player the one who can claim the useful totals after each pair. Choosing to go first is therefore not a winning strategy by itself.
A changed target gives a changed opening
Return to additions of 1, 2, or 3, but change the target to 22. Work backwards in steps of four: 22, 18, 14, 10, 6, 2. The first player can open with 2 and preserve those totals using complementary replies. Copying the opening 1 from the 21 game would miss the new pattern.
Problem
Starting at 0, what guarantees a win when the target is 22 and moves are 1–3?
- 1.Find the useful totals by subtracting four from the target: 22, 18, 14, 10, 6, 2.
- 2.The first player adds 2 to reach the earliest positive landmark.
- 3.After each opponent move, reply so that the two moves add four. The first player then announces 2, 6, 10, 14, 18, 22 and wins.
You can adapt the idea without memorising a list. If moves range from 1 up to a largest amount, add one to that largest amount to find the complementary pair total. Work backwards from the target by that pair total. If the earliest landmark is a positive amount within the allowed range, the first player can claim it. If zero is a landmark, the second player can preserve the pattern from the start.
Problem
Use moves 1–3 and a target of 20. Which player has a guaranteed strategy?
- 1.The complementary pair total is still four. Work backwards: 20, 16, 12, 8, 4, 0.
- 2.Zero is already a landmark. After the first move, the second player can add its complement to reach 4.
- 3.The second player can then announce 8, 12, 16, and 20 after later pairs. Thus the second player has the guaranteed strategy.
Create a variation by choosing a target and a largest allowed addition. Agree that the target must be reached exactly, record legal moves, and work backwards. Test the proposed strategy against every possible opponent reply. Explain why the first landmark is attainable and why the final reply is still legal.
Quiz
In the game to 21 with moves 1–3, what opening guarantees the first player control of the winning pattern?
You have reached a landmark and the opponent adds 2 in the 21 game. What is your complementary reply?
After you announce 17, the opponent announces 20. What should you add?
Who can force a win in the game to 99 with additions 1–10 and starting total 0?
What is the first player’s correct opening for target 22 with moves 1–3?
Why does complementing moves work?
Practice Problems
- Play the 21 game and record a complete game in which the first player maintains all six landmarks.
- At a landmark in the 21 game, list the three possible opponent moves and the matching replies.
- Explain how the second player can take control if the first player opens the 21 game with 3.
- In the 99 game, find the second player’s replies to moves 1, 4, 7, and 10.
- Starting at 88 in the 99 game, check the final reply for every possible move from 1 to 10.
- Find the winning player and landmarks for target 23 with moves 1–3.
- Find the winning player and landmarks for target 30 with moves 1–4.
- Design your own variation and explain a strategy that handles every legal opponent reply.
The winning totals are 1, 5, 9, 13, 17, 21. Reply to 1, 2, 3 with 3, 2, 1. If the first player opens with 3, the second adds 2 to take 5 and then preserves the four-at-a-time pattern.
Key Takeaways
• A winning strategy must cover every legal opponent reply. • Work backwards from the exact target to find useful totals. • With moves 1–3, complementary pairs total four; with moves 1–10, they total eleven. • The first player controls 1, 5, 9, 13, 17, 21 in the 21 game. • The second player controls multiples of eleven in the 99 game. • Changing the target can change the opening or which player can force a win.