Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use several base cases to support a recurrence that moves by four.
- Justify the conclusion "n=(n−4)+4, so every n≥12 is representable" using the stated assumptions.
Before you start
Induction and integer arithmetic.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Using only 4-unit and 5-unit tokens, can every integer total n≥12 be formed?
Why this math matters
Use several base cases to support a recurrence that moves by four. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Token counts are nonnegative integers.
- There is no limit on token supply.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Build a strong-induction postage argument
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Using only 4-unit and 5-unit tokens, can every integer total n≥12 be formed?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
12=4+4+4; 13=4+4+5; 14=4+5+5; 15=5+5+5
These four consecutive base cases cover the possible remainders modulo four.
Work through the mathematics
n≥16 ⇒ n−4≥12
Under the strong induction hypothesis, n−4 already has a representation.
Check the conclusion
n=(n−4)+4, so every n≥12 is representable
Adding one 4-unit token completes the inductive step without requiring negative token counts.
The result
n=(n−4)+4, so every n≥12 is representable
Adding one 4-unit token completes the inductive step without requiring negative token counts.
Common mistakes to catch
- An induction step cannot establish an unproved base case.
- A numerical list alone does not prove infinitely many totals.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Represent 23.
Show a hint
Reduce by four until reaching a base case.
Reveal answer and explanation
23=4+4+5+5+5
Starting at 15 and adding two fours reaches 23.
Practice 2
Why is checking only 12 insufficient for this step?
Show a hint
The recurrence preserves a remainder.
Reveal answer and explanation
It only seeds totals congruent to 0 modulo 4
The other three residue classes need starting representations.
Take the idea with you
Try denominations 3 and 7 and find an appropriate block of base cases.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compress integers into residue classes
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