Find patterns and logical relationships
Infer a rule from a set of cases while recognising when the evidence permits several rules.
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Identify relationships in numerical and coded information
- Test a proposed pattern against every observation
Test the full set, not one attractive pair
Compare differences, ratios, positional changes and repeated features. A code might replace symbols consistently, reverse their order or combine both operations. State a proposed rule and verify each supplied case. Unless the allowed family of rules is specified, a short sequence rarely determines a unique continuation.
Under the explicit rule ‘each letter advances one place alphabetically’, CAT becomes DBU and DOG becomes EPH. The same substitution gives MAP → NBQ. Without that rule, examples alone may support other coding systems.
Match the logical strength of the conclusion
A pattern that fits the observed cases may be a useful model, but fitting is not proof that it governs every unseen case. When a problem provides a fixed rule, apply it exactly. When you are asked to suggest a rule, mark it as one consistent possibility.
With totals or rankings, use invariants: quantities that stay fixed across cases. Check whether repeated symbols represent the same value and whether positions are allowed to change; importing an unstated convention can spoil an otherwise careful deduction.
Sources for this lesson
Pause and explain
Do three matching sequence terms always prove one unique rule?
Worked example
Encode MAP using the stated one-place alphabet shift.
Show the worked solution
- M becomes N.
- A becomes B.
- P becomes Q.
Answer MAP becomes NBQ. The result follows the supplied rule, rather than an unsupported guess about the examples.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: Encode SUN using the stated one-place alphabet shift.
Give me a hint
Replace each letter independently.
Compare my reasoning
- S becomes T.
- U becomes V.
- N becomes O.
SUN becomes TVO. Each letter uses the same one-place shift, and the order is unchanged.
Look for these in your work
- Uses a consistent shift
- Preserves order
- Checks each letter
Independent: A table gives inputs 1,2,3 and outputs 5,8,11. Within the rule family output = a×input+b, find a and b and output at 6.
Give me a hint
The output difference equals a.
Compare my reasoning
- Successive outputs rise by three, so a=3.
- At input 1, b=5−3=2.
- At input 6, 3×6+2=20.
The specified linear rule is output=3×input+2, giving 20 at input 6. It fits all three rows; uniqueness here depends on the stated linear family.
Look for these in your work
- Finds slope
- Finds intercept
- Respects the model family
Application: The observed sequence is 2,4,6. Someone says the next value must be 8. Assess the claim without an additional rule.
Give me a hint
Separate a natural suggestion from a deduction.
Compare my reasoning
- Adding two gives a possible continuation of eight.
- A rule that changes after three terms can also fit.
- No allowed-rule restriction was supplied.
Eight is a reasonable suggested continuation, but it is not forced by the three values alone. Several rules fit the observations and differ on the next term; the word ‘must’ overstates the evidence.
Look for these in your work
- Gives a fitting rule
- Recognises alternatives
- Qualifies necessity
Common mistakes
- Assuming one fitted rule is unique
- Ignoring an inconvenient example
- Changing the rule halfway through a code
What can you explain now?
Do three matching sequence terms always prove one unique rule?
Compare with the explanation
No, several rules may fit a finite sample
A specified rule family or additional constraints are needed for uniqueness.
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Make it stick
Do three matching sequence terms always prove one unique rule?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.