How four bits become a kolam

Encoding square kolam tiles, matching their edges and separating local rules from a global condition.

Take a square, mark its centre and the midpoint of each side, and choose which of the four midpoints should be connected to the centre. There are (2^4=16) choices. A four-bit word records one choice completely.

We shall read the bits in the order

[ E;N;W;S. ]

Thus (1010) has an opening to the east and west, while (0111) has openings to the north, west and south.

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

[ $$\begin{aligned} w_{1,y}&=0, & e_{4,y}&=0,\\ s_{x,1}&=0, & n_{x,4}&=0. \end{aligned}$$

]

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.

NoteContinue in the laboratory

Open the Kolam Tile Laboratory to build a configuration, inspect violated conditions and try the orbit challenge.