1×n Konane jump, capture, and add up the row
Black (Left) and White (Right) take turns jumping one of their stones over an adjacent enemy stone into the empty square beyond, removing it; a stone may keep jumping in the same direction. Whoever cannot jump loses. Every row has an exact game value, shown on the right. Pick a stone and its landing squares are marked ✓ when the jump keeps the win and ✗ when it throws it away. Dashed boxes are parts of the row that can never touch, so the value is their sum.
value of the row
Chan & Tsai (2002), rechecked
How the values are computed
A stone only ever moves two squares, so its colour is fixed by the parity of its square and a position is just the set of occupied squares. Each position is valued once from its options and reduced to canonical form, so equal games get the same id. Shifting a row by one square swaps every colour and negates the value; reversing it keeps the value; both are used to share work.
Give a stone on square x the weight φx (φ the golden ratio). A jump to the right keeps the total, because φx = φx−1 + φx−2, and a jump to the left lowers it; a finite row weighs less than one stone two squares past its end. So no stone ever gets more than one square beyond the row, and parts of the row whose reaches stay two squares apart are independent: the row is their sum. Every row of up to 13 squares gives the same value with and without this split.
Checked against the literature
The repository recomputes every cell of the eight tables in A. Chan and A. Tsai, 1×n Konane: A Summary of Results, More Games of No Chance, MSRI Publications 42 (2002) 331–339, and checks every formula and conjecture in the text on all rows up to a fixed size.