Colouring tiles. Paul Hunter. June 2010

Transcription

1 Colouring tiles Paul Hunter June Introduction We consider the following problem: For each tromino/tetromino, what are the minimum number of colours required to colour the standard tiling of the plane such that any copy of that shape is made up of different coloured tiles? We show that the minimum number of colours for each shape (see Figures 1 and 2 for notation used) is as follows: Shape Minimum number of colours I-tromino 3 L-tromino 4 I,O,S-tetromino 4 T-tetromino 5 L-tetromino 8 2 Lower bounds 2.1 I-tromino, I-tetromino, O-tetromino and S-tetromino Trivially 3 is the minimum number of colours required so that each tile of a tromino is a different colour. Likewise 4 is the minimum number of colours required for tetrominoes. I-tromino L-tromino Figure 1: The trominoes 1

2 I-tetromino O-tetromino L-tetromino T-tetromino S-tetromino Figure 2: The tetrominoes Figure 3: X-pentamino Lemma 1. At least 3 colours are required to validly colour the plane for the I-tromino and at least 4 colours are required to validly colour the plane for the I,O and S-tetrominoes. 2.2 L-tromino Suppose we can colour the plane with 3 colours so that any copy of the L- tromino consists of 3 different colours. Consider any 2 2 square of 4 tiles. By the pigeon-hole principle at least two of these tiles have the same colour and it is easy to see that a single L-tromino can cover these contradicting the validity of the colouring. Thus a valid colouring for the L-tromino must have at least 4 colours. Lemma 2. At least 4 colours are required to validly colour the plane for the L-tromino. 2.3 T-tetromino Suppose we can colour the plane with 4 colours so that any copy of the T- tetromino consists of 4 different colours. Consider any 5 tiles arranged in a cross as pictured in Figure 3. By the pigeon-hole principle at least two of these tiles have the same colour, and it is straightforward to see that any two tiles of an X-pentamino can be covered by a single T-tetromino contradicting the validity of the colouring. Thus a valid colouring for the T-tetromino must have at least 5 colours. 2

3 Lemma 3. At least 5 colours are required to validly colour the plane for the T-tetromino. 2.4 L-tetromino For the L-tetromino we need the following auxillary result. Lemma 4. If the plane can be validly coloured for the L-tetromino with 7 colours, then on any 3 3 square of 9 tiles the diagonally opposite corners (and only these pairs of tiles) have the same colour. Proof. On a 3 3 square of 9 tiles, the only pairs of tiles which cannot be covered by a single L-tetromino are the diagonally opposite corners. If the 3 3 square is coloured with 7 colours, then at least three tiles have a non-unique colour. Thus the only way for this square to be validly coloured is if both pairs of diagonally opposite corners have the same colour. Now suppose the plane is validly coloured for the L-tetromino with 7 colours and consider a 4 4 square of 16 tiles. We can partition these 16 tiles into eight pairs of diagonally opposite corners on a 3 3 sub-square. We note that any 3 3 sub-square contains at least one member of each pair. Applying the above lemma to each of the 3 3 sub-squares asserts that within each pair, both tiles have the same colour. As there are eight pairs and seven colours, by the pigeonhole principle there must be at least two pairs with the same colour. Since the 3 3 sub-square with one of those pairs as the diagonal corners contains at least one tile of the other pair, there exists an L-tetromino which covers two tiles of the same colour contradicting the validity of the colouring. Thus, Lemma 5. At least 8 colours are required to validly colour the plane for the L-tetromino. 3 Upper bounds In this section we give colourings which match the lower bounds of the previous section. For each colouring we give the minimum (non-repeating) block, the full colouring can be obtained by repeating the given colouring in the obvious way. 3.1 I-tromino and I-tetromino The colouring for the I-tromino can be seen in Figure 4 and for the I-tetromino can be seen in Figure 5. We note that in any row or column, tiles of the same colour have two (in Figure 4) or three (in Figure 5) tiles between them. As an I-tromino (I-tetromino) only covers three (four) tiles in the same row or column, it cannot possibly cover two tiles of the same colour. Thus the colourings are valid. Lemma 6. At most 3 colours are required for a valid colouring for the I-tromino and at most 4 colours are required for a valid colouring for the I-tetromino. 3

4 Figure 4: Colouring for the I-tromino Figure 5: Colouring for the I-tetromino 3.2 L-tromino, O-tetromino and S-tetromino The colouring for the L-tromino, O-tetromino and S-tetromino is shown in Figure 6. Since the L-tromino can be seen as a sub-shape of the O-tetromino, a valid colouring for the latter is also a valid colouring for the former. We note that the given colouring identifies odd and even rows and columns. That is, using a standard co-ordinate system for the tiles, each tile s colour is uniquely determined by the parity of each of its co-ordinates. Now any set of 4 tiles covered by an O or S-tetromino consists of either two adjacent tiles in one row and two adjacent tiles in the next row or two adjacent tiles in one column and two adjacent tiles in the next column (in the case of the O-tetromino, both cases occur). Thus each tile in the four has a unique pair of co-ordinates (modulo 2) and is thus coloured by a different colour. Lemma 7. At most 4 colours are required for a valid colouring of the L-tromino, O-tetromino and S-tetromino. Figure 6: Colouring for the L-tromino, O-tetromino and S-tetromino 4

5 Figure 7: Colouring for the T-tetromino 3.3 T-tetromino The colouring for the T-tetromino is shown in Figure 7. Using integer coordinates to identify tiles and the set [5] := {0,1,2,3,4} to identify colours, this colouring can be described by the function f(x,y) = 3x + y mod 5. We observe that for any x and y, f is a bijection from {(x,y),(x±1,y),(x,y±1)} to [5]. This means that for any set of 5 tiles covered by an X-pentamino, each is coloured uniquely. That is, the given colouring is a valid colouring for the X-pentamino. As a T-tetromino is a sub-shape of the X-pentamino, it follows that this is also a valid colouring for the T-tetromino. Lemma 8. At most 5 colours are required to validly colour the plane for the T-tetromino. 3.4 L-tetromino The colouring for the L-tetromino is shown in Figure 8. Using integer coordinates for tiles on the plane, we note that the Manhattan-distance (sum of difference between x-co-ordinates and difference between y-co-ordinates) between any two identically coloured tiles under this colouring is at least 4. As a single L-tetromino (indeed, any single tetromino) can only cover tiles with Manhattan-distance at most 3, there is no L-tetromino which covers two identically coloured tiles. Thus this is a valid colouring for the L-tetromino. Lemma 9. At most 8 colours are required to validly colour the plane for the L-tetromino. 5

6 Figure 8: Colouring for the L-tetromino 6