Winning strategies: working a game backwards from its end

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What to assume they know

The Collatz conjecture: a question a child can ask and nobody can settle: an argument that settles every case, as opposed to checking cases one at a time
Addition and subtraction within 100
Counting on and counting back in steps of a fixed size
Division with a remainder, at least informally

What they should be able to do

Play the game accurately, adding within the permitted range each turn
Explain why searching forwards through the possible turns is impractical
Identify the position from which the opponent cannot reach the target
Build the whole ladder of such positions by repeated subtraction
State the pattern of numbers the winning player should say, in each printed game
Decide which player wins if both play correctly, and justify the decision
Apply the same reasoning to a game with a different target and a different permitted move
Invent a variation and predict its winner before playing it
Explain why 99 and 21 behave differently under the same kind of analysis

Where it usually goes wrong

"Whoever goes first wins." Not in Game #2. Going first is an advantage only when the opener can land on the bottom rung, and the book's second game is built so that they cannot.
"You win by adding as much as you can." You win by adding whatever brings you to the next rung, which is often the smallest legal move. Greedy play loses this game reliably.
"You have to plan the whole game from the start." You have to know the ladder. After that each move is forced and takes no thought — which is precisely what makes it a strategy rather than a knack.
"Playing it many times will show you the strategy." It will show you that someone keeps winning. Finding out why means working backwards, and the chapter asks for the pattern, not the outcome.
"The same strategy works for every target." Change 21 to 22 and the winner is still the opener but their first move changes; change 21 to 20 and the opener loses outright. The target's remainder is what matters, not its size.
"Numbers in a game are decoration." The Summary names this as the chapter's example of numbers being used to play and win, and puts the naming of a set procedure at the centre of what the chapter was for.

Questions to check understanding

"Which player can always win, and why?" for a stated target and move size
"What pattern of numbers should the winner say?"
Given a position part-way through a game, state the correct move
Given a target and a permitted move range, decide the winner before play
Design a variation in which the second player wins
Explain why the winner changes when the target changes by one
Reasoning items on why playing repeatedly does not amount to finding a strategy
Items that ask for the set procedure in words — the Summary's own framing

Examples worth working on the board

Game #1 (§3.12, p.71). The opener says 1, 2 or 3. After that the two players alternate, each adding 1, 2 or 3 to the number just said. Whoever says 21 has won. The book then asks two questions: which player can force a win, and what pattern of numbers the winner should be saying.
Game #2 (§3.12, p.71). The opener says anything from 1 to 10. After that each player adds anything from 1 to 10 to the number just said. Whoever says 99 has won. The same two questions follow on p.72.
The chapter-end variation (p.73, question 10). Start from 0; players alternate adding 1, 2 or 3; whoever reaches 22 has won. The book asks what the strategy is now. Note that the target has moved by one from Game #1 while everything else is unchanged — which is exactly the right test of whether the class has the method or only the answer.
The construction, as inputs.:
- the target
- the largest amount a player may add
- the fact that a player who says the target has won Then: the gap between rungs is one more than the largest permitted addition, because whatever your opponent adds, you can top it up to exactly that gap.
Game #1's ladder. Rungs four apart, counted back from 21.
Game #2's ladder. Rungs eleven apart, counted back from 99. The lowest rung is the one the opener cannot reach, because the biggest legal opening move is 10. That single observation is the whole difference between the two games.
The deciding division. Divide the target by one more than the largest permitted addition. If something is left over, the opener wins and should say exactly that leftover on the first turn; if nothing is left over, the second player wins. Check this against all three printed games before using it — it accounts for all three, including the change of winner between the 21 game and the 22 game.
Invent your own (§3.12, p.72). The book asks the class to choose their own maximum addition and their own target, play the game, and work out who wins. The deciding division turns that from an experiment into a prediction, which is the payoff of the whole topic.

Figures to have open

A number strip from 0 to 21 on which rungs can be lit one at a time, and the same strip extended to 99 for Game #2. Standard schematic; the book prints the rules as prose and no diagram at all.
A branching diagram of the possible turns, for section 3, drawn deliberately too wide to follow. Not in the textbook, and the point is that it is unreadable.
A division frame for section 10, with target, move size and remainder as slots. Standard schematic.
No photograph, table or dataset from the textbook is required. I printed pages 71, 72 and 73 and §3.12 carries no figure of its own; the only graphic on those pages is the supercell grid belonging to question 1 of the chapter-end block, which is handled in Supercells: a number's status comes from its neighbours.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 3 "Number Play": §3.12 Games and Winning Strategies, pp.71–73
The rules of Game #1 and the two questions that follow, p.71
The rules of Game #2, p.71, with its two questions continuing onto p.72
The invitation to invent a variation, p.72
The chapter-end Figure it Out block, p.73, question 10
Summary, p.73, for the naming of set procedures and computational thinking
Backward link: §3.10, pp.68–69 (The Collatz conjecture: a question a child can ask and nobody can settle), where a question that looks equally simple has no answer at all — the pairing is the module's argument