Use and verify strategies for numerical puzzles and games

Alberta Grade 11 Mathematics

Number and money

Make a plan, use the clues, and check that your answer really works

Numerical puzzles and games have rules that limit which answers are possible. The goal is not to guess quickly. It is to use the rules, keep track of possibilities, and check the result. In a workplace or daily-life setting, the same habits help you catch a mistaken total or an incomplete plan. This lesson uses three strategies: work backward, organize possibilities, and use a number property to test a game move. A number property is a fact about numbers, such as whether a number is even or odd.

What you will learn

Choose a useful strategy for a numerical puzzle or game.
Use clues and number properties to narrow down possible answers.
Verify a proposed answer by checking it against every rule.

Start with the rules and choose a strategy

First, read the puzzle carefully. Restate each rule in your own words. Identify what is known and what must be found. If the puzzle has several clues, do not try to hold them all in your head. Write them down or mark them beside a list of possible answers.

A strategy is a planned way to make progress. Work backward when the puzzle describes steps that lead to a final result. Organize possibilities when there are only a few choices. Look for number properties when a clue refers to odd, even, or repeated results.

Before using these ideas, recall that an even whole number can be shared into two equal whole-number groups. It ends in CAD 0, 2, 4, 6, or

8

. An odd whole number does not split into two equal whole-number groups. It ends in CAD 1, 3, 5, 7, or

9

.

Translate words into clear rules before calculating.
Choose a strategy that fits the structure of the puzzle.
Record possibilities so that clues are not forgotten.

Work backward and verify each clue

When a puzzle gives a sequence of operations, reverse the operations in reverse order. Addition is undone by subtraction, and multiplication by a nonzero whole number is undone by division when the result divides evenly. For example, if a number is doubled and then increased by

7

, undo the increase before undoing the doubling.

A proposed answer is verified when you put it back into the original rules and every rule is true. This is stronger than checking only the last calculation. If a puzzle asks for a whole number, a decimal answer may fail even if some arithmetic is correct.

A quick check for reasonableness is also useful. Ask whether the answer has the required type, whether it is in the stated range, and whether its size makes sense. These checks can reveal an error, but they do not replace checking every clue.

Undo the last operation first.
Check the answer in the original puzzle, not only in your reversed steps.
Confirm that the answer meets any range or whole-number requirement.

Use organized trials and number properties

A trial is a possible answer that you test against the rules. An organized list helps you avoid repeating trials or missing a possibility. For a small set of choices, list them in order and cross out each one that breaks a clue.

Number properties can eliminate choices without testing every calculation. For example, adding two odd whole numbers gives an even whole number. If a game rule requires an odd total, two odd-numbered choices cannot make a legal move together.

In a game, separate a move that is allowed from a move that is useful. Follow the stated rules first. Then compare legal moves and explain why one helps. A strategy is not verified just because it worked once; test it against the rules and the situation described.

Use a list or table when the possible choices are limited.
Use odd and even properties to rule out impossible totals.
Distinguish a legal move from a winning or useful move.

A routine for solving puzzles

Use this routine for a puzzle or game: understand the rules, choose a strategy, carry it out carefully, and verify the result. If your answer fails one clue, do not hide that failure. Find the step that caused it, revise the plan, and check again.

The table shows how to match a puzzle feature with a starting strategy. It is a guide, not a rule that every puzzle has only one method. You can change strategies if your first attempt does not help.

A clear plan makes your reasoning easier to review.
A failed check gives information about what to revise.
Explain why a choice satisfies the rules.

Match a clue to a starting strategy

Puzzle feature	Useful first move
A sequence ends with a known result	Work backward
A short range or small set of choices	List and test possibilities
A clue mentions odd or even totals	Use odd and even properties

Worked example

Reverse a sequence of operations

A number is multiplied by

4

, then

9

is added. The final result is

41

. What was the starting number?

Undo the last step
The last operation was adding $9$ . Subtract $9$ from the final result to find the number before that addition.
$41-9=32$
Undo the multiplication
The earlier operation multiplied the starting number by $4$ . Divide by $4$ to recover the starting number.
$32\div4=8$
Verify in the original order
Put $8$ through the stated steps. It reaches the given final result, so it satisfies the puzzle.
$8\times4+9=41$

Answer: The starting number was

8

.

Check: The answer is a whole number, and the original operations produce

41

.

Worked example

Use clues to narrow a choice

A game card has one number from

12

to

20

, inclusive. The number is even, and its digits add to

8

. Find the number.

List the even choices
The range contains only a few even whole numbers. Listing them makes the range clue easy to check.
$CAD 12, 14, 16, 18, 20$
Test the digit-sum clue
Add the digits of each choice. The digit sum must be $8$ , so choices with a different sum are removed.
$1+2=3,\quad1+4=5,\quad1+6=7,\quad1+8=9,\quad2+0=2$
Check whether a choice remains
None of the listed even numbers has digits that add to $8$ . Therefore the clues do not describe any number in the stated range. A good solver reports the conflict instead of inventing an answer.
$\varnothing$

Answer: There is no number that satisfies all three clues.

Check: The list includes every even number from

12

through

20

, and none has a digit sum of

8

.

Worked example

Test a number-game move

In a game, a player chooses two different numbers from CAD 1, 2, 3, 4, 5 and adds them. A move is legal only if the total is odd. Is choosing

2

and

5

legal? Give another legal pair and verify it.

Check the proposed pair
Add the two chosen numbers. The result must be odd for the move to be legal.
$2+5=7$
Find another possible pair
Choose one even number and one odd number. For example, $1$ is odd and $4$ is even. Their sum is odd.
$1+4=5$
Verify the game rules
Both pairs use different numbers from the allowed set, and both totals are odd. Therefore both moves are legal. This does not show that either move guarantees a win; the stated rule only tests whether a move is legal. 7 is odd,\quad5 is odd

Answer: Choosing

2

and

5

is legal. Another legal pair is

1

and

4

.

Check: All four numbers used are in the allowed set, and each pair has an odd total.

Common mistakes and how to avoid them

Undoing the operations in the same order they were done.

Correction: Start with the last operation and reverse the order.

Stopping after finding a possible answer.

Correction: Substitute the answer into the original rules and check every clue.

Changing a clue when no answer appears to fit.

Correction: Report that the clues conflict, then show the possibilities you checked.

Calling a game move a winning strategy just because it is legal.

Correction: Use only the conclusion supported by the rules. Legal does not necessarily mean winning.

Lesson summary

Read and record the rules before choosing a method.
Work backward by undoing operations in reverse order.
Organize a small set of possibilities and test each clue.
Use odd and even properties when they apply.
Verify the final answer against every original rule.

Check your understanding

Question 1

A number is increased by

6

, then doubled to make

30

. What was the number?

$9$
$12$
$15$
$18$

Show answer and explanation

9

Undo doubling first:

30\div2=15

. Then undo adding

6

:

15-6=9

. Checking gives

(9+6)\times2=30

.

Question 2

Which pair has an odd total?

$2$ and $4$
$3$ and $5$
$1$ and $6$
$4$ and $8$

Show answer and explanation

1

and

6

1+6=7

, which is odd. The other totals are

6

,

8

, and

12

, all even.

Question 3

A puzzle says the answer is an even number from

5

to

9

, and its digits add to

7

. What is the answer?

$6$
$7$
$8$
No answer fits

Show answer and explanation

7

The range contains even numbers

6

and

8

, but the digit-sum clue fits neither. The number

7

has digit sum

7

but is not even. Thus no answer fits all clues.

Key terms

Strategy: A planned method for making progress on a puzzle or game.
Trial: A possible answer that is tested against the puzzle rules.
Verify: Check that an answer satisfies the original rules.
Even number: A whole number divisible into two equal whole-number groups.
Odd number: A whole number that cannot be divided into two equal whole-number groups.

Continue through Math 20-3

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-3), expectation N1. 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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