Weekly Problem: Number Theory

The Collatz Sequence

Let’s explore one of the most deceptively simple yet unsolved problems in mathematics, first proposed by Lothar Collatz in 1937.

The Rules

Start with any positive integer n. At each step:

If the number is even, divide it by 2
If the number is odd, multiply it by 3 and add 1
Repeat this process with the resulting number

Example 1: Starting with n = 6

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Example 2: Starting with n = 7

7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

Once the sequence reaches 1, it enters the cycle: 1 → 4 → 2 → 1

This Week’s Problems

1. Exploration

Choose five different starting numbers between 10 and 20. For each number:

Write out the complete sequence until you reach 1
Count how many steps it takes to reach 1
Find the highest number that appears in your sequence

2. Pattern Finding

Based on your calculations:

Do larger starting numbers always take more steps to reach 1?
Do numbers that are close to each other take a similar number of steps?
What happens with powers of 2 (like 2, 4, 8, 16)?

3. Challenge Question

Find a starting number less than 100 that takes more than 15 steps to reach 1. Prove that your answer is correct by showing the complete sequence.

Historical Context

This problem has been studied extensively by mathematicians for over 80 years. Despite its simple formulation:

No one has proven that all starting numbers eventually reach 1
Computers have verified the conjecture for all numbers up to 2⁶⁸
The sequence can reach numbers much larger than the starting number before eventually decreasing

Notable Case: Starting with n = 27

27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → 137 → 412 → 206 → 103 → 310 → 155 → 466 → 233 → 700 → 350 → 175 → 526 → 263 → 790 → 395 → 1186 → 593 → 1780 → 890 → 445 → 1336 → 668 → 334 → 167 → 502 → 251 → 754 → 377 → 1132 → 566 → 283 → 850 → 425 → 1276 → 638 → 319 → 958 → 479 → 1438 → 719 → 2158 → 1079 → 3238 → 1619 → 4858 → 2429 → 7288 → 3644 → 1822 → 911 → 2734 → 1367 → 4102 → 2051 → 6154 → 3077 → 9232 → 4616 → 2308 → 1154 → 577 → 1732 → 866 → 433 → 1300 → 650 → 325 → 976 → 488 → 244 → 122 → 61 → 184 → 92 → 46 → 23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1