Practice · Number Theory · Collatz
Collatz Sequence Practice Problems
Practice problems on the Collatz conjecture (3n+1): work out stopping times and peak values, judge true/false claims. Rising difficulty plus a boss, free.
Q1 of 6
0 correct
How many steps does 10 take to reach 1?
n = 10
Quick answer
What's the best way to practice the Collatz sequence?
Run several starting numbers all the way through: if the number is even, halve it (n/2); if it is odd, compute 3n + 1. Write each sequence out with arrows and count the arrows — that is the stopping time. At the same time, note the largest value that appears (the peak value). Mix the tasks: sometimes find the step count, sometimes judge a claim like "27 takes more than 100 steps" as true or false. Example: 10 → 5 → 16 → 8 → 4 → 2 → 1 is 6 steps.
HowTo
A 4-step solution strategy
This order works for any positive whole starting number.
- 1Step 1 of 4
Write the sequence out cleanly
Start with the starting number and write each new value behind an arrow: n → … → 1. That way you lose nothing while counting and can check your work later.
- 2Step 2 of 4
Apply the rule per step
Even number → divide by 2. Odd number → multiply by 3 and add 1 (3n + 1). Never apply both rules to the same number.
- 3Step 3 of 4
Read off stopping time or peak value
The stopping time is the number of arrows down to 1. The peak value is the largest number anywhere in the sequence — note them separately.
- 4Step 4 of 4
Compare deliberately for claims
If a claim needs checking ("stopping time > 100?"), compute the sequence up to the point in question and compare the result with the claim.
Examples
Worked examples with full working
Four Collatz sequences step by step. Try each one yourself first, then compare.
Easy
How many steps does 6 take?
6 is even → 6/2 = 3
3 is odd → 3·3+1 = 10
10 → 5 → 16 → 8 → 4 → 2 → 1
Count arrows: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
8 arrows, peak value 16
The stopping time of 6 is 8 steps.
Medium
What peak value does 7 reach?
7 → 22 → 11 → 34 → 17 → 52
52 → 26 → 13 → 40 → 20 → 10
10 → 5 → 16 → 8 → 4 → 2 → 1
Mark the largest number that appears
Peak value 52, stopping time 16
Even though 7 is small, the sequence climbs to 52 before it falls.
Medium
How many steps does 11 take?
11 → 34 → 17 → 52 → 26 → 13
13 → 40 → 20 → 10 → 5 → 16
16 → 8 → 4 → 2 → 1
Count all the arrows
14 arrows, peak value 52
Stopping time 14 — typically chaotic compared with its neighbors.
Hard
Boss: How many steps does 18 take?
18 → 9 → 28 → 14 → 7 → 22
22 → 11 → 34 → 17 → 52 → 26
26 → 13 → 40 → 20 → 10 → 5
5 → 16 → 8 → 4 → 2 → 1
20 arrows, peak value 52
18 starts even, then runs on through 9 and 7 for a long time — stopping time 20.
Pitfalls
Common mistakes — and how to avoid them
These five traps keep coming up when practicing Collatz.
Rule swapped
Even → halve, odd → 3n + 1. Flip those and you get an entirely different sequence and a wrong stopping time.
Counting the starting number
The stopping time counts steps (arrows), not numbers. In 16 → 8 → 4 → 2 → 1 there are 5 numbers but only 4 steps.
Confusing stopping time and peak value
The stopping time is the step count, the peak value is the largest number reached. For 27: 111 steps, but peak value 9232.
Forgetting the +1 in the 3n+1 step
An odd number means 3·n + 1, not just 3·n. 7 becomes 22, not 21 — that +1 decides the whole rest of the sequence.
Expecting a steady decline
The sequence does not fall steadily. It can climb sharply (7 up to 52) before it finally reaches 1 — don't stop too early.
Study
Practice with a plan — three quick tips
By hand first, calculator second
Work each sequence by hand with arrows first, then compare with the Collatz calculator. That way you spot mistakes exactly where they happen.
Note stopping time and peak value separately
Keep two columns: one for the step count, one for the largest number. Then you won't mix up the two quantities in an exam.
Compare neighboring numbers
Work neighboring starting numbers like 26 and 27 side by side (10 vs. 111 steps). That vividly shows how chaotically stopping times jump around.
FAQ
Frequently asked practice questions
Glossary
Terms in one sentence
- Collatz conjecture
- The conjecture that the (3n+1) sequence ends at 1 from every positive starting number.
- Stopping time
- The number of steps until the sequence first reaches 1.
- Peak value
- The largest number the sequence reaches along the way.
- (3n+1) rule
- For an odd number: multiply by 3 and add 1.
- Iteration
- Repeatedly applying the same rule to each new result.
- Cycle 4 → 2 → 1
- The final loop that every Collatz sequence ends in.
- Boss question
- The last and hardest task of the practice set, here the longest sequence.