How this instrument works
The rule behind the Collatz Conjecture could hardly be simpler to state: take any positive whole number, and if it's even, cut it in half; if it's odd, triple it and add one. Repeat that on whatever comes out, again and again, and every case anyone has ever tried by hand or by computer eventually drops down to 1. Six becomes 3, becomes 10, becomes 5, becomes 16, becomes 8, becomes 4, becomes 2, becomes 1 — eight steps from a small start to the same destination every known sequence shares.
What makes this genuinely famous is the gap between that observation and an actual proof of it. Researchers have verified the claim by computer across numbers into the hundreds of quintillions without a single counterexample turning up, yet nobody has shown it must hold for every positive whole number, however large. It sits among a short list of problems simple enough for a child to try and unsolved by everyone who has ever attempted a general argument, which is exactly why it keeps drawing fresh attention decades after it was first posed.
This page deliberately shows one step only, not the full chain down to 1. A complete sequence has no known upper bound on how long it might run before finishing, since that bound is precisely the unproven part of the conjecture — a tool built to 'run until it reaches 1' would be quietly assuming the very claim nobody has established. Entering a number here and reading back the single next value keeps this sheet honest about what is actually known versus merely observed across trillions of trials.
- Enter your starting whole number into Current number (n).
- Read Next number in the Collatz sequence: half of n if n is even, or three times n plus one if n is odd.
- Feed that returned value back into Current number (n) by hand to trace several steps yourself.
- Watch for a landing on 1 — the sequence then cycles 4, 2, 1 forever, which is why reaching 1 counts as finished.
- Try a large odd starting number to see how sharply a single step can climb before any downward trend shows up.
Worked example — one step from 6, 7, and 1
Set Current number (n) to 6. Since 6 is even, the rule halves it: 6 ⁄ 2 = 3, so Next number in the Collatz sequence reads 3. Set n to 7 instead, an odd number, and the rule triples it and adds one: 3 × 7 + 1 = 22, a jump upward rather than down. Odd inputs behave this way generally, climbing first before any later step brings them back toward smaller values.
Set n to 1, the sequence's own eventual destination, and one step still runs the odd rule: 3 × 1 + 1 = 4. That is not a glitch — from 4 the sequence would continue 4 to 2 to 1 to 4 again, circling the same three numbers forever, which is exactly why reaching 1 the first time is treated as the sequence being finished rather than fed back in over and over.
Questions
What does this calculator actually compute?
Exactly one step of the Collatz rule applied to whatever number is entered: half of it if the number is even, or three times the number plus one if it's odd. It does not chase a full sequence down to 1, since how many steps that would take is not something mathematics currently guarantees for every starting number.
What is the Collatz Conjecture itself claiming?
That repeating this halve-or-triple-and-add-one step, starting from any positive whole number, always eventually reaches 1. Every number checked by computer so far, into the hundreds of quintillions, does reach 1, but no proof exists that this holds for literally every positive whole number, which is why it remains a conjecture rather than a settled theorem.
Why is this considered such a famous unsolved problem?
Because the rule is simple enough to explain to a child yet has resisted proof from professional mathematicians across roughly a century of attempts. That combination, trivial to state and stubbornly hard to settle, is rare, and it accounts for much of why the conjecture keeps attracting new attempts and partial results.
Why does this sheet stop after a single step?
Because no known bound exists on how many steps a full sequence might take before reaching 1, so a tool that ran automatically until it hit 1 would be leaning on the unproven part of the conjecture itself. Showing one step at a time keeps things honest: only what is actually computed appears, never an assumed ending.
What happens once a sequence reaches 1?
It cycles. One step from 1 gives 4, one step from 4 gives 2, one step from 2 gives 1, and those same three values repeat endlessly after that. Reaching 1 is treated as the finish line precisely because continuing past it only ever produces that same three-number loop.
Does every odd number produce a bigger next value?
Yes, always. Tripling any positive number and adding one produces something larger than the number you started with, so a single step from an odd input is guaranteed to climb. Whether the overall chain trends downward across many steps is a separate question, one the conjecture answers only in its unproven, general form.