Dawson's chess
octal .137Computing from .137.
Normal play, finite proof, open tail
The Pattern That Has Not Arrived
Dawson's chess repeats forever after a finite certificate checks 89 equalities. Grundy's game, its unequal-splitting cousin, has been computed through at least 284,944,236,543 heaps without a known period. Run both nim-value sequences from their rules, watch one lock and the other refuse every tested period, then follow the sparse values on which the open question turns.
The two-channel recurrence bench
Run the same question twice
Each cell is a heap size. Each row is 34 heaps wide. Cell hue records the residue modulo 34; brightness records the nim-value computed by mex.
Grundy's game
unequal split, dot=sparseA direct 500-heap frame is ready.
0Dawson heaps generated
500Grundy heaps generated
computingsparse occurrences in prefix
computinglargest Grundy nim-value in prefix
The Dawson channel certifies immediately. The longer channel streams in a worker.
A nim-value is not a score. It is the one Nim heap equivalent to a game position. For any legal move, XOR the nim-values of the heaps left behind; the current heap receives the mex, the smallest non-negative integer missing from those options.
Octal code .137 gives three removal rules. Remove one only when that empties the heap; remove two and leave zero or one heap; remove three and leave zero, one, or two heaps. The page executes those digits directly. This is Guy and Smith's normal-play analysis. Dawson's original pawn problem used the opposite last-move convention, so the two results must not be conflated.
A finite door into an infinite tail
The Guy-Smith certificate
For an octal game whose last nonzero digit is at position k, one finite equality window is enough:
if G(n+p)=G(n) for n0 <= n < 2*n0+p+k, then G(n+p)=G(n) for every n >= n0
Here the page uses n0=52, p=34, and k=3. Its verifier re-derives the browser-prefix nim-values and checks the full Guy-Smith window. The upper endpoint is 2*52+34+3=141, excluded, so the test contains exactly 0 equalities.
Computing exceptions to the eventual residue template.
Running certificate.
Looking periodic is not the same as being periodic
Dismissal one
"A bounded sequence from a finite rule must cycle."
The premise is missing. Grundy's nim-values are not known to be bounded. The cited computation reaches heap sizes above two billion while new record nim-values still appear. A finite rule does not supply bounded state for free.
Dismissal two
"Either it looks periodic or you have not gone far enough."
The Dawson channel answers this exactly. Its 89 passing equalities do not make a persuasive picture; the theorem converts them into a proof for every later heap. The Grundy channel answers more modestly: every tested period fails, which is evidence about this prefix and nothing more.
The deliberately limited negative scan
Every candidate from 1 through 512 fails
The page applies the same equality-window shape at n0=8192, with a displayed comparison parameter k=3. This is not a Guy-Smith certificate for Grundy's game, because unequal splitting cannot be expressed by standard octal notation. A passing window would not prove its tail. A failed window does conclusively reject that candidate on the stated prefix.
Waiting for the computed prefix.
The scale objection deserves one more datum. Flammenkamp's computational table reports octal .106 settling in 2002 with period 328,226,140,474 after preperiod 465,384,263,797. Astronomical periods can be found and certified. This does not decide Grundy's game. It makes a run past 2.8*10^11 terms more than casual eyeballing.
The thin trickle
Delete the lowest bit of a Grundy nim-value and count the one-bits that remain. Even parity marks a sparse value; odd parity marks a common value. XOR preserves this parity relation, so the split is not decoration. It is an exact way to partition the mex search.
Sparse value
G=10 = 10102
Drop the low bit: 101 has two ones. Even parity.
Common value
G=8 = 10002
Drop the low bit: 100 has one one. Odd parity.
Flammenkamp's self-published computational record reports 1,287 sparse occurrences through the 2012 run, with the last known one at G(48,399,022)=259. The page reproduces only the first million terms and prints its own smaller count above. The deeper implication is conditional and exact: if sparse occurrences stop, the infinite sequence must become periodic. The open question can therefore be watched as a thinning trickle. No source establishes whether it ends.
The check
Recomputed here
Dawson fixture, exception set, finite certificate, and computed tail:
Grundy fixture, direct recurrence cross-check, finite scan, and sparse prefix:
Cited observations
Flammenkamp's 2012 endpoint, 1,287-record sparse ledger, last known sparse occurrence, and `.106` period data are not reproduced by this browser run.
Free choices and limits
Browser horizon: 1,000,003. Period scan: p=1..512, n0=8192, comparison parameter k=3. Grid width: 34. These choices diagnose a prefix; none proves Grundy periodic or aperiodic.
| Layer | What the page may conclude | What it may not conclude |
|---|---|---|
| Observed record | The cited 2012 run reached at least its stated endpoint without a known period. | That this is the latest possible computation by anyone. |
| Diagnostic inference | No period through 512 passes the declared Grundy prefix window. | That Grundy's sequence is aperiodic, bounded, or unbounded. |
| Theorem | The passing Dawson window certifies period 34 for every n from 52 onward. | That the same theorem applies to the non-octal unequal-split rule. |
How the million-term computation avoids a quadratic wall
The naive rule tests every unequal split. The live worker instead uses the sparse/common parity identity. Sparse XOR common is common; common XOR common and sparse XOR sparse are sparse. It first visits every split with a sparse side, which finds every reachable common value. Once the first missing common value is known, mex cannot exceed it. A second pass checks only the still missing sparse candidates below that ceiling. This partitions the complete move set without changing it. The offline verifier checks the accelerated result term by term against the direct recurrence through n=4096.