Null Space Catch

Null Space Catch is a navigation puzzle built on one idea from linear algebra: the null space of a matrix is the set of all vectors that the matrix sends to zero. A probe sits at the left edge of a dark canvas. A gold target pulses at the right. Between them: coloured zones, each containing a 2×2 matrix. In each zone, the probe can only move in directions that the zone's matrix sends to zero. The matrix panel below the canvas shows the matrix, the computed null space direction, and the verification that the matrix times that direction equals zero. This is not an approximation or a metaphor. The movement constraint is the null space, computed directly from the matrix entries. For a matrix with first row (1, 2), the null space satisfies x + 2y = 0, which gives the direction (−2, 1). The probe moves in that direction — diagonally up-left or down-right — and in no other. A different matrix gives a different direction. The zones are colour-coded and filled with faint parallel lines running along their null space directions, so the allowed movement is readable visually as well as mathematically. The puzzle is geometric. The probe's path through n zones is a sequence of line segments, each at the angle determined by that zone's null space. The player must arrive at each zone boundary at precisely the correct vertical position so that the next zone's null space direction carries them toward the target. Arriving one centimetre too high or too low causes the path to miss the target entirely, requiring backtracking. Three difficulty levels. The first uses three zones with axis-aligned and simple diagonal null spaces. The second uses four zones where the path bends twice, requiring the player to plan two entry positions simultaneously. The third uses five zones including one that requires backtracking — moving backward along the null space before continuing forward — to position correctly for the next zone's entry. The scoring rewards path efficiency. A probe that moves directly through each zone scores more than one that overshoots and corrects. The theoretically minimum path — no backtracking, perfect entry at each boundary — gives the maximum score. Finding that path requires understanding that the null space direction determines not just where the probe can go, but exactly where it will be at every zone boundary given any starting position. The game teaches what the null space of a matrix actually is, geometrically, by making it the thing you navigate through.