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LR1 · Analyze numerical and logical puzzles or games using problem-solving strategies

Learn to analyze numerical and logical puzzles or games using problem-solving strategies through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Logical Reasoning

Organize clues, test possibilities, and plan game moves

Puzzles often seem difficult because several rules must be considered at once. A useful strategy is to make the information visible, test possibilities in an organized way, and check the result against every rule. This lesson focuses on those habits for numerical and logical puzzles or games. The goal is not to guess quickly. It is to make each step clear enough that another person can follow it.

What you will learn

Represent puzzle information with clear variables, lists, or tables.
Use organized testing and logical deduction to reduce possible answers.
Explain why a strategy works and check that a proposed answer meets every condition.

1. Review: organize the information first

A numerical puzzle uses numbers and conditions. A logical puzzle uses clues that tell you what can or cannot be true. A solution is a value or arrangement that satisfies all the conditions.

Before solving, read the whole puzzle once. Then list what is known, what is unknown, and what rules must be obeyed. If a puzzle has several positions, label them. If it has a small number of possibilities, write them down rather than trying to remember them.

A variable is a letter used to stand for an unknown number or choice. For example, you might call an unknown number

n

. A condition such as “the number is even” is then a rule that

n

must satisfy. Use only as much symbolic notation as helps make the puzzle clearer.

A table is useful when the puzzle involves repeated tests or several cases. Include headings that show what is being tested and what happens. Keep track of ruled-out cases too; otherwise, you may repeat work or overlook a possibility.

Separate clues from guesses.
Label unknown quantities and positions.
Record tests so the reasoning stays organized.

2. Choose a strategy and use it carefully

Start with the most restrictive clue: the clue that eliminates the most possibilities. For example, a clue that fixes a number’s last digit may be more useful than a clue that only says the number is greater than a small value.

When there are only a few possible cases, use an organized list. Test each candidate against the clues in the same order. Cross out a candidate as soon as it fails a rule. If every candidate is tested, the remaining answer or answers are justified rather than guessed.

For a pattern puzzle, compare nearby terms. Ask what changes from one term to the next. A difference is the amount added or subtracted to move from one term to the next. Check whether the differences repeat or follow a simple pattern. Do not assume that a few matching terms prove a rule; use the puzzle’s stated conditions and test the rule on all given terms.

For a game, identify the legal moves and the condition for winning. A legal move is one the rules allow. Consider a few moves from the end of the game and work backward. If a move gives the other player a position from which they cannot force a win, it may be a strong choice. State the assumptions: for example, whether both players know the rules and choose strategically.

A complete solution has three parts: a method, a result, and a check. The check is important because a calculation can be correct while the answer still violates a clue.

Use the strongest clue early.
Test cases in a consistent order.
For games, include the legal moves and win condition in your reasoning.
Check the final result against every rule.

3. Make reasoning visible

A small table can make a repeated test easier to follow. Suppose you want to know whether a number can be the sum of two different whole numbers from 1 through 6. List the smaller number first in each pair. This avoids testing the same pair twice in reverse order.

A number puzzle may also be represented with an equation. An equation is a statement that two quantities are equal. If two different whole numbers add to 9, record the condition as

a+b=9

, then check the allowed values of

a

and

b

. The equation records the rule; it does not replace the need to check that the values are allowed and different.

When you explain a solution, connect each step to a clue or rule. “I chose this because it looked right” is not enough. Instead, say which possibilities were removed and why.

a+b=9

Tables help compare cases without losing track.
Equations can record numerical conditions.
A clear explanation links each decision to a puzzle rule.

4. Practice, then check

Before solving, decide what form of record will help: a list for a few candidates, a table for repeated tests, or a short equation for a numerical relationship. Avoid doing extra calculations that do not narrow the possibilities.

After solving, read the original puzzle again. Check the number of answers requested, the allowed values, and every logical condition. If there may be more than one answer, show why the other possibilities fail or explain why the puzzle permits several solutions.

In a game, test your strategy against the rules, not just one imagined round. A choice that wins in one situation may fail in another. Describe the position or condition in which your strategy is meant to work.

Choose a representation that fits the puzzle.
Check completeness as well as correctness.
Do not claim a game strategy works beyond the cases you have considered.

Tracking changes in a number pattern

Term	Value	Change from previous term
1	2	—
2	5	3
3	10	5
4	17	7
5	26	9

Worked example

Find a number from clues

A two-digit number has different digits. Its digits add to 11, and the tens digit is 3 greater than the units digit. Find the number.

Name the digits
Let $t$ be the tens digit and $u$ be the units digit. Each digit must be a whole number from 0 through 9. Since this is a two-digit number, $t$ cannot be 0.
Translate the clues
The digit-sum clue says the digits add to 11. The tens digit being 3 greater means that adding 3 to the units digit gives the tens digit.
$t+u=11,\quad t=u+3$
Find the digits
Substitute the expression for $t$ into the digit-sum condition. The result gives $u=4$ , so the tens digit is $7$ . These digits are different and both are valid digits.
$u+3+u=11,\quad u=4,\quad t=7$
Write and check the number
The tens digit comes first, so the number is 74. Its digits are different, add to 11, and the tens digit is 3 greater than the units digit.

Answer: The number is 74.

Check: The clue equations give

t=u+3

and

t+u=11

. They produce

u=4

and

t=7

. The digits are valid and distinct.

Worked example

Use differences to extend a pattern

A game board displays the sequence 2, 5, 10, 17. The next value is formed by the same rule. Find the next value and explain the rule.

Compare consecutive values
Subtract each term from the term after it. This makes the changes visible instead of relying on a visual guess.
$5-2=3,\quad 10-5=5,\quad 17-10=7$
Look for a pattern in the changes
The amounts added are 3, 5, and 7. Each is 2 greater than the previous amount, so the next amount in this pattern is 9.
Extend and check
Add 9 to the last value, 17. The next value is 26. Check that the new change continues the stated pattern.
$17+9=26$

Answer: The next value is 26, using the rule of adding consecutive odd numbers beginning with 3.

Check: The differences are 3, 5, 7, and then 9. This fits the stated continuation rule. A finite list can sometimes support more than one possible rule, so the answer uses the simple consistent pattern intended by the puzzle.

Worked example

Reason backward in a token game

Two players take turns removing 1 or 2 tokens from a pile of 7. The player who removes the last token wins. If you move first and both players follow the best strategy, how many tokens should you take?

Start near the end
With 1 or 2 tokens left, the player whose turn it is can take all the tokens and win. With 3 tokens left, taking 1 leaves 2 and taking 2 leaves 1. Either move lets the other player take the remaining tokens and win.
Identify a useful position
A player who starts a turn with 3 tokens cannot avoid giving the opponent the last token. With 6 tokens, taking 1 leaves 5 or taking 2 leaves 4. From 5, the opponent can take 2 and leave 3; from 4, the opponent can take 1 and leave 3. Thus, 6 is a position from which a player can leave 3 for the opponent.
Choose the first move
Take 1 token from the pile of 7. This leaves 6 for the other player. Whatever they do, you can respond so the two turns remove 3 tokens in total: if they take 1, take 2; if they take 2, take 1. Your response leaves 3 for the other player.
$7-1=6,\quad 1+2=3$
Check the final position
After you leave 3, the opponent must take 1 or 2. You can then take the remaining 2 or 1 and remove the last token. This checks both possible opponent moves.

Answer: Take 1 token first. Then respond to each opponent move by taking enough that the two turns remove 3 tokens in total.

Check: After your first move, 6 tokens remain. If the opponent takes 1, you take 2 and leave 3; if the opponent takes 2, you take 1 and leave 3. From 3, the opponent takes 1 or 2, and you take the remaining tokens to win.

Common mistakes and how to avoid them

Trying possibilities in a random order and forgetting which ones were tested.

Correction: Use a short, organized list or table, and test each case consistently.

Stopping when an answer fits one clue.

Correction: Check the answer against every clue and every restriction in the puzzle.

Assuming a number pattern has only one possible continuation.

Correction: State the rule you are using and check that it matches every given term.

Calling a game move a winning strategy without checking the opponent’s replies.

Correction: List the opponent’s legal responses and test what your next move would be.

Lesson summary

Translate clues into clearly labelled conditions.
Use lists, tables, or equations to keep cases organized.
For patterns, compare terms and state the rule used to continue them.
For games, reason from the winning condition and test the opponent’s legal replies.
Finish by checking every clue and confirming that the conclusion follows.

Check your understanding

Question 1

A number is greater than 20 and less than 30. Its digits add to 8. What is the number?

Show answer and explanation

The number must be from 21 through 29. Of those numbers, only 26 has digits that add to 8.

Question 2

What is the next term in 4, 9, 16, 25 if the differences increase by 2 each time?

Show answer and explanation

The differences are 5, 7, and 9. Since each difference increases by 2, the next difference is 11. Then

25+11=36

Question 3

In a token game, a player may remove 1 or 2 tokens, and taking the last token wins. Why should a proposed move be checked against the opponent’s replies?

Because the opponent may have more legal moves than the first player.
Because a move that looks good may allow the opponent to reach a winning position.
Because every game must end after exactly two turns.
Because the number of tokens always increases after a move.

Show answer and explanation

Because a move that looks good may allow the opponent to reach a winning position.

A strategy must account for the opponent’s available responses. A move is not shown to be strong just because it removes tokens.

Key terms

Candidate: A possible answer that has not yet been ruled out.
Condition: A rule or clue that a solution must satisfy.
Difference: The amount found by subtracting one value from another; in a sequence, it describes the change between consecutive terms.
Legal move: A move that follows the stated rules of a game.
Variable: A letter used to represent an unknown number or choice.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation LR1. 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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