Proof: direct, contrapositive, contradiction and induction

You want to prove “if $n^2$ is odd, then $n$ is odd” by contrapositive. What do you assume at the start?

Use Gauss's formula to compute $1 + 2 + 3 + \dots + 200$.

In the “all horses are the same colour” induction, where exactly does the argument break down?

Let $N = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 = 30031$. What does Euclid's argument guarantee about $N$?

With an unlimited supply of stamps worth $3$ and $7$, what is the largest amount that cannot be made exactly?

Which of these statements is proved most naturally by strong induction rather than ordinary induction?

“Let $a = b$. Then $a^2 = ab$, so $a^2 - b^2 = ab - b^2$, so $(a+b)(a-b) = b(a-b)$, so $a + b = b$, so $2b = b$, so $2 = 1$.” Which step is invalid?

A student proves: “if $n^2 + n + 1$ is even, then $(n+1)^2 + (n+1) + 1$ is even”, and concludes by induction that $n^2 + n + 1$ is even for all $n$. Which statement is correct?

Which Lean proposition expresses the principle of ordinary induction for the natural numbers?

Write is_prime(n) and first_failure(claim, limit), which returns the least n with 0 <= n <= limit for which claim(n) is false, or None if there is none. Use them to find the first n for which $n^2 + n + 41$ is not prime. Why does your search not prove anything about the values beyond limit?