THE HACKENBUSH — a dart

THE HACKENBUSH cut the picture, read the number

A drawing of colored edges rooted to the ground; players take turns cutting edges (Left cuts blue, Right cuts red), and anything no longer connected to the ground falls away. The last to move wins. The miracle: every red-blue Hackenbush position equals a surreal number — a pile of blue edges is +n, a blue-then-red stalk is ½. The game IS its value; positive means Left wins.

THE TECHNIQUE position value = a surreal number

The demo reads the value of an all-blue stalk (Left’s advantage) and a balanced position (a zero game): live demo

HISTORY & CREDIT Berlekamp, Conway, Guy

“Hackenbush is just a counting puzzle.” — its blue-red positions realize the whole dyadic surreal line; a blue-red stalk can equal ½, ¼, any dyadic rational. cited

cut & fall · each player cuts their color; disconnected edges drop.
the value · all-blue stalk = +n; a value > 0 means Left wins regardless of who starts.
Winning Ways · Berlekamp, Conway & Guy — the game-to-number bridge (with [[the-surreal-numbers]]).

A picture that is secretly a number. combinatorial-game

RECOMMEND FOR I-13 the value, on the compiler

On i-13, a 3-edge all-blue stalk has value +3 (Left wins), a balanced blue/red position is a zero game:

$ i13 run gm_hackenbush.i13 RUN OK · 28 step(s) value = 3 left_wins = 1 -- all-blue-3 = +3 bval = 0 balanced_is_zero_game = 1 -- 2nd player wins

Recommend as a NULL — a pinned valuation (B39); seated axis. The Hackenbush value is fixed by the position, and it is an instance of the game→surreal-number bridge ([[the-surreal-numbers]]), not a new invariant. NULL — cut the picture, read the number.