The pigeon who built a square staircase
Pip adds one odd-shaped border each day. Mika discovers a square hiding in the stairs.
The formula in this story
Story 8 of 8
What you will take away
Explain why a starting case and a general next step are both needed.
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What do the words mean?
An odd positive integer is 2k + 1 for a nonnegative integer k. The first n positive odd integers end at 2n − 1, where n ≥ 1 is an integer.
Want a hand with the prerequisites?The missing two cake strips
A suspicious plan
Day one: Pip owns one tile. Day two: he finds three. “My staircase is hungry.”
Start with one tile: 1 = 1². Every staircase needs a first step.
The clue
Mika fits the three tiles around the first: a 2 by 2 square. Five more make a 3 by 3 square. Pip suspects excellent interior design.
For the general next step, add 2n + 1 tiles around an n-by-n square. The two arms share one corner.
The twist
Each new L-shaped border has two arms and one corner. The staircase was a growing square all along.
A starting case and a repeatable next step justify every n. Changing n shows examples, not a replacement for that reasoning.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Start at n = 1: the sum is 1 = 1².
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For every positive integer n, the sum of the first n positive odd integers is n².
- Start at n = 1: the sum is 1 = 1².
- Suppose the first n odd numbers total n². The next odd number is 2n + 1. Adding it gives n² + 2n + 1 = (n + 1)².
- The starting case and this repeatable step prove every positive integer case. This is mathematical induction: not just several checked examples.
Pip has one more idea…
Does checking n = 1, 2 and 3 alone prove the formula for all n?
Read the explanation for Yes, three examples establish the pattern.
Patterns suggest a claim. The induction step is what extends this claim beyond the examples.
Read the explanation for No, we also need a reason that every next step works.
Yes. We supplied both a base case and a general next step, so the argument reaches every positive integer.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
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These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.