From bits to a drawing

The straight segments are the combinatorial skeleton. In the kolam rendering, the same edge data is drawn as a smooth white strand on a coloured square. The visual style changes; the information does not.

Write the tile in column (x), row (y) as

[ t_{x,y}=(e_{x,y},n_{x,y},w_{x,y},s_{x,y}). ]

Two horizontally adjacent tiles match precisely when

[ e_{x,y}=w_{x+1,y}, ]

and two vertically adjacent tiles match when

[ n_{x,y}=s_{x,y+1}. ]

These are local equations: each one looks at only a single shared edge.

Closing the boundary

For a (4) board, no strand should leave the outer square. With rows numbered upwards, this says

]

The matching and boundary equations guarantee that the small pieces join without loose ends. They do not guarantee a single kolam: the board may contain several disjoint loops.

The global condition

Make a graph whose vertices are tile centres and shared active edges. The complete drawing is connected exactly when this graph has one non-trivial connected component.

This distinction is useful:

• boundary closure is local to the outside edge;
• edge matching is local to neighbouring squares;
• connectedness is global and may depend on the entire board.

An arrangement can satisfy every local equation and still fail the global test.

Why turn it into a 15-puzzle?

Set aside the (0000) tile, regard its position as the empty square, and allow only the usual slide into that square. During a slide sequence, intermediate boards need not satisfy the kolam conditions. We can then ask a different question:

Which valid kolam configurations can be reached from which others by legal 15-puzzle moves?

This introduces a classical parity invariant, but now the target arrangements come from a highly structured family. The interactive laboratory lets you investigate the construction problem and the sliding problem separately before putting them together.