🧠 Part A: Questions About the Nature of the Method

1. What is the real role of the base step? If it is missing, what could go wrong? Are there statements for which the base step "does not really matter"? Explain.
2. In the inductive hypothesis we do not know whether the statement is true. How is it still "legal" to use it in order to prove the statement for \(k+1\)?
3. Can the base step use \(n=5\) instead of \(n=1\)? What would change? Give examples of statements for which this is possible, and statements for which it is not.
4. If the step from \(k\) to \(k+1\) is valid — does that guarantee the step from \(k+1\) to \(k+2\) is also valid? Explain why or why not.
5. What is the difference between "checking a few cases" and proving by induction? Why does our brain love examples — and why is that dangerous?

📐 Part B: Questions About the Inductive Step

6. What kind of use of the inductive hypothesis is considered "legal"? Present an example of a valid use and an example of an invalid use.
7. Why can we not simply "assume" that the left-hand and right-hand sides are approaching each other? Think about common student errors.
8. Why do we often add \(k+1\) to both sides? Show an example where this works well, and an example where it does not help at all.
9. What happens when the inductive step depends not only on \(k\) but also on intermediate values? Does induction still work?
10. What is the difference between a "step formula" and an "expansion" of the expression? Is explicit substitution always necessary?

🔶 Part C: Deep Geometric Questions

11. Suppose a triangle is divided into \(n\) strips parallel to the base. We saw that the number of small triangles is \(n^2\). Can you create a version of "triangle multiplication"? What would happen if you replaced "strips" with "points"? Investigate the idea.
12. Explain how geometric induction works even for non-uniform decompositions. What conditions must the geometric structure satisfy for induction to be applicable?
13. Can you invent a geometric shape for which induction does not work? Justify your answer based on the regularity of the inductive step.

💬 Part D: Induction on Processes – Creative Thinking

Induction is not limited to algebraic statements. It can also be applied to processes.

14. A child builds a tower of cubes: at each step a new row of cubes is added in a "staircase" shape — one more cube than the previous row. Formulate an appropriate statement and propose a proof by induction.
15. A game: at each step one ball is added to a jar, and every 3 steps two balls are removed. Can you investigate the "number of balls" using induction? What is the condition for the inductive step?
16. A sorting process: numbers are inserted into a list one by one, and at each step one small change is made to the list. How can you formulate a statement on which induction is performed?

🌀 Part E: The Induction Mindset – Thinking About Thinking

17. In which situations does induction feel "natural", and in which does it feel forced? Give examples.
18. Why does induction work only on well-ordered structures? What is the connection between "natural numbers" and the possibility of applying induction?
19. What is the logic behind the statement: "To prove infinitely many cases – it is enough to prove two" (the base + the step). Explain in your own words.