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Cyborg Mathematics

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Combinatorial games

An octal game is a two-player game of removing counters from heaps, with rules coded as a short string of digits. Whether such a game settles into a repeating pattern is unknown for many of them, so people compute the values as far as they can. Achim Flammenkamp keeps the reference tables.

2 results · 1 article

Results

Numbered by their place in the strongest-first list

  1. 03

    The take-away game 0.007, computed four times past its published table

    In a take-away game, two players take turns removing counters from heaps under a fixed rule, and a table of values says which moves win. In July 2026 my run computed that table for the game 0.007, known as Treblecross, for every heap size up to ▲, four times further than the table that had stood since 2021. Achim Flammenkamp, who keeps the reference tables, checked the program himself and reran the game over his old table’s range with his own programs, getting the same results, and he has since taken the game further with his own runs.

    Checked outside this site

    • RerunAchim Flammenkamp checked the program and reran the game over his old range, . his octal tables
    • OEIS editorPublished as a new OEIS entry, . OEIS A395248
    SequencesGames
  2. 04

    More take-away games extended, three of them rerun by the tables’ maintainer

    In July 2026 I extended the published tables of more of these games, ▲ in all counting 0.007, two to four times further than before. Flammenkamp reran three of them, 0.006, 0.606 and 0.114, over their full new ranges on his own machines with his own programs and got the same answers; with 0.007, he reran four of the games in all. In game 0.114 the runs also found more heap sizes with rare values. Rare values are the exceptions in a game’s table; if they ever stop appearing, the table is known to repeat forever, which is how such a game gets solved.

    Checked outside this site

    • RerunAchim Flammenkamp reran three of the games with his own programs. his octal tables
    • CreditHis tables added the extensions with credit by name, . the credit line
    • OEIS editorThe extended OEIS entries were published, the last of them on this date, . OEIS A071459
    SequencesGames

Articles

Notes on the numbers marked ▲

  • Heap sizes covered by my July 2026 run of the game 0.007 (about 1.07 billion): 2^30. This was four times the table that had stood since 2021, and the furthest published computation of the game when it finished in July 2026. Achim Flammenkamp's tables have since gone further: to 2^31, about 2.1 billion heap sizes, by September 2026 (his own run), and to 2^34, about 17 billion, as listed on his page (checked October 29, 2026). The furthest values are now on his page.
  • Unsolved take-away games whose published tables I extended in July 2026: 12. Achim Flammenkamp, who keeps the reference tables, has since taken every one of these games further, so his page (checked October 29, 2026) now has the furthest values.