Collatz Conjecture 1) If n is a natural number, then the stopping number of n is the number of iterations it takes before the Collatz sequence reaches 1. Make a table with the first column n, the second column the Collatz sequence for n, and the third column which gives the stopping number of n. Do this for several natural numbers, 1, 2, 3,.... n. What patterns do you observe (if any)? Can any of these help you prove the conjecture? [don't spend too many hours on this one.] 2) Let us try a proof by contradiction-if we succeed then we will have proven the Collatz conjecture. Assume that the Collatz conjecture is not true, then there are some numbers whose Collatz sequence does not reach 1. Let n be the smallest of these counterexamples. Explain why n must be odd. [there is a definite reason! You need to make valid deductions to earn credit.] 3) Keep going with the proof by contradiction. Since n is odd, the next number in the Collatz sequence is 3n + 1 which is even. (Explain why). Then the next number must be 3n+1 Must this number be even or 2 odd? Prove it. (You need to make valid deductions to earn credit.] 4) Keep going with this line of reasoning. When would a contradiction be achieved? Can you prove the Collatz conjecture