Plausible reasoning, live
An agent doing number theory.
It reads integer sequences from the OEIS one term at a time, writes a rule down the moment one fits, and checks it against every published term it hasn’t seen.
The worksheet · live
Opening…
3Carry out the planCan you check each step?
2Devise a planHave you seen it before?
pinned down still right ruled out
4Look backCan you check the result?
When the checking is done it files every rule that held in its library, and if too many of its recent guesses were refuted, it asks for more terms before the next one.
Its working
01Pólya’s four steps
George Pólya’s How to Solve It (1945) breaks solving a problem into four steps. The agent takes them for every sequence, and the worksheet above shows which one it is on.
- Understand the problem
It takes a sequence from the OEIS: ten famous ones first, then any of the first 400,800 at random, with readers’ sequences jumping the queue. It keeps only the name, the offset and the listed terms. The formulas and programs people wrote are dropped on arrival, so it can’t peek. Then it reads the terms one at a time.
- Devise a plan
From the fourth term on, it first asks whether it has seen the pattern before, trying the rules in its library. Then it tries every shape of rule it knows, simplest first, fitted exactly in rational numbers. A rule only counts if it fits with equations to spare.
- Carry out the plan
The first rule that fits becomes a conjecture. It is sealed in the record before the agent sees another term. Only then does it download the b-file, the OEIS’s long list of published terms, and check up to a thousand of them. One miss refutes it, and it reads on and tries again, up to five times.
- Look back
A rule that held goes into the library. If more than 45% of its last 20 conjectures were refuted, it asks for more evidence before the next one; if fewer than 20% were, it relaxes.
The rules it knows
Rule (3) covers the Catalan and Motzkin numbers, factorials and central binomial coefficients. A rule must fit with at least two more equations than it has unknowns, more for bigger rules and more again when the agent has grown cautious, so it can’t simply memorise the terms.
What it can’t do
“No conjecture” means none of (1)–(3) fits, not that there is no rule: the primes, the partition numbers and most of the OEIS follow rules it can’t express. A conjecture that holds agrees with every published term. That is evidence, not proof.
02Results
The table fills in as the agent works.
| Sequence | Conjecture | After | Outcome |
|---|
03What it has learned
It learns from its own record, and only between sequences, so each sequence is read with one fixed state of mind. Every twenty minutes it writes what happened in its notebook.
A library of rules
A library rule has nothing left to fit, so it needs fewer agreeing terms: five for a rule that has held once, down to three for one that has held on three or more sequences, and one more for every time it has fooled the agent.
| Rule | Held on | Terms checked |
|---|
How cautious it is
04Send it a sequence
Give it an OEIS number and the sequence goes to the front of the queue. Your name goes on its conjectures in the record.
05The record
Every conjecture is written down before the agent compares it with a single term it hasn’t read, and every outcome after. Each entry carries the hash of the one before it, so nothing can be changed or quietly removed. Checking the chain… The whole chain is at /api/ledger.
| No. | Written (UTC) | Entry | Sequence | Hash |
|---|