Chapter 11 · Logical Reasoning II

Match Numbers to Roles and Arithmetic Clues

Lesson 11.2 · 数字角色与算术线索

A number is not useful until you know what it represents. Learn to match candidates to years, months, days, ages, sums, differences, products, and quotients—then audit every clue.

Meaning first. Arithmetic second. Check every candidate once.

Grade 5 enrichmentSelf-contained10 missionsObjective checks

Mission 1

A blank has a role, not just a number

Before calculating, ask what kind of quantity belongs in each blank.

Not complete

📅

YearUsually a four-digit number in these puzzles.

🗓️

MonthA whole number from 1 through 12.

☀️

Day of monthA whole number from 1 through the last day of the given month; 31 is the largest possible day number.

🎂

Age or durationA count of completed or explicitly defined years. Calendar-year difference is not always completed age: if this year’s birthday has not happened, subtract one more year.

Important: A range only tells you what is possible. It does not always prove the final role. For example, 3 could be a month or a day until other clues are used.

In these puzzles, four-digit candidates represent years. Which role does 1955 have?

Which value cannot be a month or day?

A number that fits a range is…

What should stay attached to every candidate?

Mission 2

Use the strongest clue first

Some candidates have only one possible role; others need arithmetic or one-use-only reasoning.

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Candidate set

1955318797618

Inspect one candidate (the age/duration range in this model is 0–130 years)

Place low-ambiguity candidates first

1955 and 1879 are the only four-digit choices, so they fill the two year roles.

76 is too large for a month or day, so it fills the age role.

Then use what remains

After the year and age roles are filled, the remaining 3 and 18 must fill month and day.

Because 18 cannot be a month, month = 3 and day = 18.

How many four-digit candidates are there?

Which candidate is forced into the age role by range?

After those roles are filled, which candidate must be the month?

What principle prevents reusing one candidate?

Mission 3

Worked example: place the Einstein numbers

All biographical information needed for this number-placement puzzle is included here.

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Complete puzzle statement

The worked example describes Albert Einstein as born in Ulm, Germany, on the 14th day of month B in year A. It says he died in the United States on April D in year C, and gives age entry E.

Place these five candidates into A, B, C, D, and E, using each exactly once:

1955318797618

A · Birth yearFour-digit year before the death year.

B · Birth monthMonth number from 1 to 12.

C · Death yearFour-digit year after the birth year.

D · Death dayDay number from 1 to 31.

E · Age entryThe difference between the two year candidates.

Mission 4

Use arithmetic to connect the roles

The two years and the age are not three separate guesses. They must satisfy one equation.

Not complete

Later year C

−

Earlier year A

Difference76

The difference 76 appears in the candidate list, so the three roles support one another.

Arithmetic clue

1955 − 1879 = 76

This confirms the order of the year candidates and the age entry.

Full audit

Every role has one candidate.
Every candidate is used once.
Month and day are in valid ranges.
The year difference matches the age entry.

For the Einstein puzzle, death year minus birth year

Which equation is the useful arithmetic clue?

Why check every candidate once?

What is the final step?

Mission 5

Extra practice: place the Beethoven numbers

The same role structure appears with a different candidate set.

Not complete

Complete practice statement

The worked example describes Beethoven as born near Bonn, Germany, on the 16th day of month B in year A. It says he died in Vienna on March D in year C, and uses age entry E.

Place these candidates into A, B, C, D, and E, using each exactly once:

26571827121770

original convention: Treat this as a number-placement puzzle. The worked example uses the simple year difference 1827 − 1770 = 57 for the age entry; no outside biography research is needed.

A · Birth year

B · Birth month

C · Death year

D · Death day

E · original age entry

Mission 6

Transfer the method to a new timeline

This added teaching puzzle uses the same role-and-arithmetic structure in a non-biographical setting.

Not complete

New museum timeline puzzle

A science museum opened in year A, month B. It completed a major renovation in year C and held a celebration on day D of that month. The calendar-year difference, renovation year minus opening year, is E.

Use each candidate once:

2012820241512

OpeningMonth B, Year A

C − A = E

Renovation celebrationDay D, Year C

A · Opening year

B · Opening month

C · Renovation year

D · Celebration day

E · Years between

Best reason for placing 12 as E instead of the month

Mission 7

original exercise: match cards to arithmetic roles

Now the roles are not year, month, day, and age. They are sum, difference, product, and quotient.

Not complete

Complete card puzzle

There is exactly one card for each number 1 through 9. Players A, B, C, and D each take two cards; no card can belong to two players. The order of the two cards within a pair does not matter.

A's two cards have sum 10.
B's larger card minus the smaller card equals 1.
C's two cards have product 24.
D's larger card divided by the smaller card equals 3.

One card remains. Find it.

A · sum 10

Choose two cards

B · difference 1

Choose two cards

C · product 24

Choose two cards

D · quotient 3

Choose two cards

Remaining card

Used by nobody

Mission 8

Enumerate candidates instead of guessing

Systematic enumeration is organized proof: list candidates, combine them without repetition, and keep only complete assignments.

Not complete

A · sum 10

B · difference 1

C · product 24

D · quotient 3

Strategy: begin with the clue having the fewest candidate pairs, but still test every branch that remains.

A sum 10	B diff 1	C product 24	D quotient 3	Left
Run the search after you have attempted Mission 7.

Which clue has the fewest candidate pairs?

How many complete non-repeating assignments survive?

Which card remains?

What makes enumeration a proof?

Mission 9

Independent workshop

Solve at least 6 of 8 fresh problems. Each has a local hint.

Not complete

1. Use 2006, 9, 2023, 28, 17 once for opening year, opening month, later year, celebration day, and year difference. What is the opening year?

Hint

Identify the two years and put them in time order.

2. For that same candidate set, what is the year difference?

Hint

Subtract the earlier year from the later year.

3. For that same candidate set, which number is the month?

Hint

After assigning years and duration, which remaining number fits 1–12?

4. For that same candidate set, which number is the day?

Hint

A month cannot be 28.

5. Separate question about a new candidate 8: does being between 1 and 12 alone prove that it belongs in a month slot?

Hint

A candidate can fit more than one role.

6. Cards numbered 1–8: how many unordered pairs of different cards have product 12?

Hint

List factors with both cards in range.

7. Cards numbered 1–6: A has a pair summing to 7; B has a pair with product 8; no card repeats. B must take 2 and 4. What is the smaller card in A’s pair?

Hint

Remove B’s cards before testing sum pairs.

8. A person is born in December 2000. In March 2020, is their completed age 20?

Hint

Calendar-year difference is not always completed age.

Mission 10

Independent exit ticket

Solve all five fresh problems. The certificate also requires Missions 1–9.

Not complete

1. Use 2008, 6, 2022, 25, 14 once for opening year, month, later year, day, and year difference. What is the later year?

Hint

The later year is the larger of the two four-digit candidates.

2. For that candidate set, what is the duration?

Hint

Use the arithmetic relationship between the years.

3. For that candidate set, what is the month?

Hint

Remove the years and duration, then use the month range.

4. Cards 1–6: A’s pair sums to 9; B’s pair has product 10. No card repeats. What is the smaller card in A’s pair?

Hint

Find B’s product pair first. Then test A’s sum pairs without reusing either of B’s cards.

5. Why is A = {4,5}, B = {2,5} not valid in that card puzzle?

Hint

A local arithmetic check is not a full assignment audit.

Achievement unlocked

Number-Role Logic Architect

This certifies that the learner can classify number roles, use arithmetic relationships, enforce one-use-only assignments, and enumerate arithmetic clues systematically.

Lesson 11.2 · Grade 5 Math Reasoning

Optional reflection — not automatically graded

Which clue type helped you most: range, digit length, arithmetic relationship, or one-use-only elimination? Explain with one example.

Teaching notes

The museum timeline, role probe, search table, workshop, and assessments are added instructional scaffolds.The Beethoven activity deliberately follows the original's simple year-difference convention for its age entry and is presented as a number-placement exercise rather than an outside historical-fact quiz.