A GAME BY CRISTOFFER CRONE
Make a world.
Make it difficult.
Two squares make a country. Two minds shape the map.
Select two empty squares to make country 0.
OPENINGS & SOLVED CONTINUATIONS
The Gomulda book
Explore a line, analyse a position, and keep what has been learned.
Loading positions…
The initial value has not been proved.
Initial position · White to pair
Unsolved
All values are White’s colouring penalty: White seeks more, Black seeks less. Studying a line here does not change your match.
Loading and checking the book…
Completed work is saved on this device while the page is open. Exports let you keep or transfer the book. Imports replace this device’s book after verification.
Choose Gomulda Search + book in a new match to use it. Original Search and Zero remain separate. A stronger selected level triggers fresh search when a stored estimate was made with less thinking time.
Continuations
| Next pair · click to study | White’s score | Evidence |
|---|
What counts as solved?
Search estimates guide exploration. Exact final colourings establish facts. A position is marked proven only when rigorous lower and upper bounds meet, accounting for every distinct legal reply that matters. Already formed countries also give a guaranteed minimum penalty. A solved continuation can establish a bound for its parent; one good-looking line cannot prove an opening.
The book merges rotations, reflections, country renaming and different pairing orders. General graph equivalence is not used yet. Imported labels are ignored: final colourings and proof implications are recomputed.
A SEPARATE LEARNING ENGINE
Gomulda Zero
Learn through self-play or practise against Gomulda Search in both roles.
A neural network learns which pairs to explore and how costly a position may become. Its search generates new games; their exact colouring penalties become training targets.
White pairs first and tries to make the final colouring expensive. Black pairs second and tries to keep it cheap. The network receives the current role at every position, including after the players swap sides.
How the role is encoded
A dedicated input is +1 for White and −1 for Black. Training values are +penalty ÷ 10 for White and −penalty ÷ 10 for Black. Search maximizes White’s penalty and minimizes it for Black. The player’s seat number does not determine this input.
Training runs on this device while this page is open. Completed training is saved here. More games or lower training loss do not, on their own, establish stronger play.
Loading learning checkpoint…
Chart: value prediction loss during training. Playing strength is measured separately below.
The test plays eight paired matches from held-out openings, with equal time per move. Each engine gets both roles. Test games never update the network or enter its training data.
Strength test
No local test yet. Gomulda Search remains the default.
How this differs from a large AlphaZero project
This is a compact policy/value network with PUCT search, legal-move masking, exploration noise, replay, board-symmetry augmentation and gradient updates. It is a working research engine, not a claim of expert or superhuman strength. A larger model and much longer, carefully evaluated training may be needed.
Self-play retains the AlphaZero-style loop. Training against fixed Search is an opponent-training variation: Zero learns from its own decisions and the outcomes of those games, without copying Search’s moves.
Colouring stays exact in both engines. Gomulda Zero learns the pairing phase. The model uses the fixed game rules, including 25 points per square for the sixth colour.
Background: DeepMind’s AlphaZero implementation notes ↗As White, make the countries
hard to colour cheaply.
As Black, keep the map friendly.
Then find its cheapest colouring.
Play again with roles reversed.
The higher White score wins.
ORIGINS & RECOLLECTIONS
The story of Gomulda
Personal recollections by Søren Riis
A name for many games
My friend Cristoffer Crone invented Gomulda around 1974. He invented many original games, and several shared the name Gomulda, including games known as 5 × 5 Gomulda and Elastik Gomulda.
The country-pairing and colouring game played here became the most popular of these inventions. Its players shared one board, with White trying to make the countries difficult to colour and Black trying to make them easier.
Mathematics and university life
When I was still in high school, Cristoffer was working on his master’s thesis. He was a fantastic source of mathematical knowledge for me.
I enjoyed parties at Elers’ Kollegium and Regensen, two prestigious student residences in Copenhagen. I later lived at Regensen myself.
Cristoffer gave me the confidence to occasionally skip high school, take the train to Copenhagen, and sit in on lectures at the University of Copenhagen. Meanwhile, the red marks and warnings for my absences accumulated at school.
Dansk Gomulda Laug
Together with Cristoffer, we founded Dansk Gomulda Laug, a small association of about 20 members. We played the game and maintained our own ratings.
I recall that my rating, around 93, was by far the highest. The other players’ ratings ranged from the low 60s to the high 70s. Curiously, our beloved Cristoffer, the game’s inventor, always had one of the lowest ratings.
Laug means “guild” or “association” in English.
These are my personal memories. The invention date and university starting year are approximate; I have yet to confirm Cristoffer’s birth and death dates.
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