How Often a Random Catan Board Puts 6 Next to 8

The result

Take the 19-hex base board, put the desert anywhere, and deal the 18 number tokens at random. Repeat 200,000 times. In 68.9% of the layouts at least one pair of neighbouring hexes carried a 6 and an 8.

Mean pairs per layout: 0.99. The raw counts are in random-boards.csv.

How we ran it

• Board: the standard 19-hex layout (rows of 3-4-5-4-3, axial coordinates, six neighbours per interior hex).
• Tokens: the real multiset — one each of 2 and 12, two each of 3, 4, 5, 6, 8, 9, 10, 11 — dealt onto the 18 non-desert hexes.
• Desert: uniformly random among the 19 positions, because variable setup shuffles the terrain too.
• Runs: 200,000 layouts from seed 7, so the counts above reproduce exactly. The script is deal.py (CPython, ~3 s).

The whole experiment is short enough to re-run in a browser console or a Python script. The core loop:

for _ in range(200000): deal(tokens, desert=randrange(19)) hits = sum(1 for a,b in edges if {num[a],num[b]} == {6,8}) counts[min(hits,5)] += 1

Why one pair matters

A 6 and an 8 each appear on 5 of the 36 dice combinations. An intersection touching both collects on 10 of 36 rolls — 27.8% of the game, before any other hex is counted. That is why competitive setups and most house rules forbid the two red numbers from touching, and why a board that breaks the rule tends to produce one runaway player.

The same logic is behind the balanced board rules: red numbers apart, duplicates apart, resources spread. Our score prices a touching 6/8 pair at −18 points, the heaviest single penalty in the formula.

What about the letters on the back?

Official variable setup does not deal the tokens freely: the tokens are lettered on the back and placed in alphabetical order along a spiral, which is a deliberate anti-clustering device. That procedure is not the random deal measured above, so 68.9% is the number for a shuffled pile, not for the rulebook method.

We have not published a figure for the lettered method yet. Modelling it requires choosing a spiral convention, and the answer moves a lot with that choice — so the honest next step is a physical test: lay out one real set twenty times and count. Until then, treat 68.9% as the baseline that applies whenever players shuffle the tokens by hand, which is what most tables actually do.

How the generator handles it

With No adjacent 6 & 8 switched on (the default), HexBoardGen rejects every candidate that breaks the rule and keeps the best of 30 that survive. Across 4,000 generated boards the count of adjacent 6/8 pairs was 0 — by construction, not by luck.

Generate a board with the rule on · the token counts used here

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