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NL2 · Analyze spatial puzzles and games with problem-solving strategies

Learn to analyze spatial puzzles and games with problem-solving strategies through clear examples and targeted practice.

Alberta Grade 11 Mathematics

Number and Logic

Use a clear plan to understand positions, test moves, and explain why a solution works

A spatial puzzle asks you to reason about positions, shapes, or moves. A game adds rules that limit which moves are allowed. Solving one is not just a matter of guessing. You can make progress by understanding the goal, recording the current situation, choosing a useful strategy, and checking each move. This lesson uses small original puzzles to practise those habits. No special formula is needed; careful, organized reasoning is the main tool.

What you will learn

Describe the goal and rules of a spatial puzzle or game before trying moves.
Use diagrams, labels, and organized records to keep track of positions.
Choose and test a strategy, then adjust it using what you learn.
Explain why a solution works and check that it follows every rule.

Review: positions, rules, and records

A position is the arrangement of pieces or objects at one moment. A move changes that arrangement. A rule tells you which moves are allowed or what must be true at the end. Read the rules before you begin. If even one rule is missed, a move that seems helpful may not be legal.

For a grid puzzle, name rows and columns so each square has a clear location. For example, a square in row 2 and column 3 can be recorded as (2, 3). A diagram makes the layout visible, while a written record helps you compare one position with another. Use a consistent convention: do not switch the row and column order halfway through.

A strategy is a planned way to make progress. Examples include trying a move and recording its result, looking for a repeated pattern, working backward from the goal, or splitting a large task into smaller tasks. These are general problem-solving tools, not guarantees that every puzzle has a solution.

Know the goal and every rule.
Label positions consistently.
Record moves so you can review them.

A practical problem-solving cycle

First, restate the goal in your own words. A goal should be specific enough that you can tell when it has been reached. “Make the board better” is unclear. “Move the token to the marked square without entering a blocked square” is clear.

Next, describe the starting position and the choices you are allowed to make. Then choose a strategy that fits the puzzle. If there are only a few possible moves, test them one at a time. If the same kind of position appears again, look for a pattern. If the goal gives strong clues about the last move, consider working backward.

After a move, check two things: was it allowed, and did it help? A move can be legal but unhelpful. If you reach a position you have already recorded, compare the available choices. Repeating the same choices may lead you around in a loop rather than toward the goal.

When a trial fails, keep the information it gives you. Note what blocked progress and change one part of your plan. Changing several things at once makes it harder to tell which change mattered. A good explanation describes the plan and the evidence that supports it, not only the final answer.

Understand the goal before choosing a move.
Test moves systematically and record results.
Use unsuccessful attempts to improve the plan.

Strategies for spatial puzzles and games

A diagram is useful when the puzzle depends on where pieces are. Mark blocked locations, legal moves, and the goal square. For a board game, a small position record may be enough. Keep the diagram simple; it should help you see the puzzle, not decorate it.

Working backward means starting from the goal and asking what position could come just before it. This can reduce the number of choices you need to test. It is especially useful when the final move is tightly restricted. You must still check that the resulting moves can be made in the forward direction.

Breaking a task into smaller parts means solving one manageable part at a time. For example, first move a piece out of a crowded area, then move it toward the destination. Check that completing the first part does not make the next part impossible.

A pattern is a repeated feature that can guide later moves. Do not assume a pattern continues just because it appeared once. Compare enough positions to see what stays the same and what changes. A useful strategy also includes a stopping check: confirm that the final position meets the goal and obeys all the rules.

Choose a strategy that matches the puzzle.
Working backward and splitting tasks can reduce confusion.
Check patterns rather than assuming they continue.

Explain and check a solution

A complete solution has three parts: the moves or decisions, the reason for choosing them, and a check against the rules. A list of moves without reasons may be hard to follow. A claim that a plan works without checking the final position is incomplete.

Use exact language. Say which piece moves, where it starts, and where it ends. If a rule limits movement, explain how the move follows that rule. If a puzzle asks for a particular arrangement, compare your final arrangement with that target.

For a game, distinguish between a move that is allowed and a move that is likely to help. You may not be able to prove that a move guarantees a win. Be honest about what your testing shows. For instance, you can say that a move avoids a repeated position in the cases you tested, but do not claim it always wins unless your reasoning supports that claim.

A reasonableness check asks whether the answer makes sense in context. Did you use only legal moves? Did you accidentally move a piece twice in one turn? Does your record match the diagram? Can another person follow your explanation and verify the result?

Show the moves and the reasons behind them.
Verify the final arrangement against every rule.
Do not claim more than your reasoning establishes.

Worked example

Find a route through a small grid

A token starts at (1, 1) on a 3-by-3 grid. It may move one square up or one square right at a time. The square (2, 2) is blocked. Find a legal route to (3, 3).

Describe the choices
The token can increase its row by one or increase its column by one. It cannot enter the blocked square. Recording each position will make the route easy to check.
Try a route
Move right first, then move up twice, and then move right once. This route avoids the blocked square and reaches the target. (1,1)\to(1,2)\to(2,2)
Correct the route
The trial written in the previous line would enter the blocked square, so it is not legal. Use the other order: move up twice, then move right twice. Each step changes exactly one coordinate by one, and none enters (2, 2). (1,1)\to(2,1)\to(3,1)\to(3,2)\to(3,3)

Answer: One legal route is (1, 1), (2, 1), (3, 1), (3, 2), (3, 3).

Check: Every move is one square up or right. The route never visits (2, 2), and it ends at (3, 3).

Worked example

Work backward from a required final move

A game uses three counters in a row. On a turn, a player must move exactly one counter one place to the right. Counters cannot pass one another or occupy the same place. The counters begin at positions 1, 2, and 3. Can they reach positions 2, 4, and 6?

Check the total movement
Each move increases the position of exactly one counter by one. The first counter must move one place, the second two places, and the third three places. The total is six moves, so the target is not ruled out by the move count.
$(2-1)+(4-2)+(6-3)=6$
Test the order of moves
Move the third counter from 3 to 4, then the second from 2 to 3, and then the third from 4 to 5. The counters are now at 1, 3, and 5. Move the first counter to 2, the second to 4, and the third to 6. No counter passes another or shares a place. (1,2,3)\to(1,2,4)\to(1,3,4)\to(1,3,5)\to(2,3,5)\to(2,4,5)\to(2,4,6)
Check the target
The final positions match the requested target, and each transition moves only one counter one place to the right. The order of moves matters because moving a counter into an occupied place is not allowed.

Answer: Yes. The six moves shown reach positions 2, 4, and 6 legally.

Check: There are six one-place moves, matching the total increase in the three positions. The recorded sequence also confirms that no counter passes another or lands on an occupied place.

Worked example

Plan moves around a blocked square

A token begins at (1, 1) on a 4-by-4 grid and must reach (4, 4). It may move one square up or right. The squares (2, 1) and (3, 2) are blocked. Give a route and explain how you know it is legal.

Notice the first restriction
The token cannot move up from (1, 1), because (2, 1) is blocked. Its first move must be right to (1, 2). This is an example of using a restriction to narrow the choices. (1,1)\to(1,2)
Avoid the second restriction
From (1, 2), moving right to (1, 3) keeps the token away from (3, 2). Then move up to row 2 and row 3 in column 3. The blocked square at (3, 2) is in a different column. (1,2)\to(1,3)\to(2,3)\to(3,3)
Finish and verify
Move right to (3, 4), then up to (4, 4). Each move is one square in an allowed direction. The route avoids both blocked squares and reaches the goal. (3,3)\to(3,4)\to(4,4)

Answer: A route is (1, 1), (1, 2), (1, 3), (2, 3), (3, 3), (3, 4), (4, 4).

Check: The route uses only right and up moves. It does not visit (2, 1) or (3, 2), and its last position is the target.

Common mistakes and how to avoid them

Trying moves without recording them.

Correction: Write down each position or draw it. A record helps you notice repeated positions and check whether a move was legal.

Assuming a move is useful because it is allowed.

Correction: After each move, ask whether it brings you closer to the goal or creates a problem for later.

Changing several parts of a plan after a failed attempt.

Correction: Keep what you learned and change one part at a time. This makes it clearer why the next attempt works or fails.

Claiming a strategy always works after testing only one case.

Correction: State what your evidence shows. Test more positions before making a broader claim.

Lesson summary

Restate the goal and list the rules before making a move.
Use a labelled diagram or written record to track positions.
Choose a strategy such as working backward, testing moves, or splitting the task into parts.
Check that every move is legal and that the final position meets the goal.
Explain the reasoning, including what you learned from unsuccessful attempts.

Check your understanding

Question 1

A token moves one square right or up. It is at (2, 1), and (2, 2) is blocked. Which statement is correct?

It must move up because that is the only legal move.
It can move right to (2, 2).
It can move up to (3, 1), but it cannot move right to (2, 2).
It can move down to (1, 1).

Show answer and explanation

It can move up to (3, 1), but it cannot move right to (2, 2).

Moving up increases the row to 3 and avoids the blocked square. Moving right would enter (2, 2), and moving down is not one of the allowed moves.

Question 2

You reach a position you have seen before in a game. What is the most useful next step?

Assume the game is won.
Compare the available moves and your earlier record for that position.
Erase the earlier record so you can start fresh.
Make any move without checking the rules.

Show answer and explanation

Compare the available moves and your earlier record for that position.

Comparing repeated positions can reveal a loop or show which choices remain. The record helps you decide based on evidence.

Question 3

What should a complete puzzle explanation include?

Only the final position.
A list of moves, a reason for the plan, and a check against the rules.
A claim that the strategy always works, even if only one case was tested.
Every possible move, whether legal or not.

Show answer and explanation

A list of moves, a reason for the plan, and a check against the rules.

A complete explanation gives the moves, explains the reasoning, and verifies that the result follows the rules.

Key terms

Position: The arrangement or location of pieces at a particular moment.
Move: An allowed action that changes a position.
Strategy: A planned way to make progress toward a goal.
Work backward: Start with the goal and reason about what could happen immediately before it.
Pattern: A feature that repeats or changes in a consistent way across positions.

Continue through Math 20-2

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation NL2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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