T Madas What is a Polyomino T Madas

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© T Madas

© T Madas

What is a Polyomino? © T Madas

What is a Polyomino? © T Madas

What is a Polyomino? • Monomino • Domino • Triomino • Tetromino • Pentomino

What is a Polyomino? • Monomino • Domino • Triomino • Tetromino • Pentomino • Hexomino • Heptomino • Octomino etc It is a shape made up of touching squares © T Madas

What is a Polyomino? It can have a hole It is a shape made

What is a Polyomino? It can have a hole It is a shape made up of touching squares • Monomino • Domino • Triomino • Tetromino • Pentomino • Hexomino • Heptomino • Octomino etc Full edge to edge contact only © T Madas

Clearly there is only 1 monomino There is only 1 domino Polyominoes produced by

Clearly there is only 1 monomino There is only 1 domino Polyominoes produced by rotations & reflections do not count as different shapes. = © T Madas

Clearly there is only 1 monomino There is only 1 domino Polyominoes produced by

Clearly there is only 1 monomino There is only 1 domino Polyominoes produced by rotations & reflections do not count as different shapes. © T Madas

Clearly there is only 1 monomino There is only 1 domino There are 2

Clearly there is only 1 monomino There is only 1 domino There are 2 triominoes There are 5 tetrominoes We need to be organised if we are to find all the pentominoes © T Madas

3 2 1 4 5 7 6 8 9 12 10 11 there are

3 2 1 4 5 7 6 8 9 12 10 11 there are 12 pentominoes © T Madas

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Which pentominoes have line symmetry? Which pentominoes have rotational symmetry and of what order?

Which pentominoes have line symmetry? Which pentominoes have rotational symmetry and of what order? Which pentominoes could be the net of an open top cubical box? Every pentomino has an area of 5 square units but do they all have the same perimeter? The 12 pentominoes have a total area of 60 square units. By tessellating all 12 pentominoes is it possible to make up: a 6 x 10 rectangle a 5 x 12 rectangle a 4 x 15 rectangle © T Madas

Pentominoes with reflective symmetry © T Madas

Pentominoes with reflective symmetry © T Madas

Pentominoes with rotational symmetry order 2 order 4 order 2 © T Madas

Pentominoes with rotational symmetry order 2 order 4 order 2 © T Madas

Pentominoes which fold to an open top box Shading their bases © T Madas

Pentominoes which fold to an open top box Shading their bases © T Madas

The perimeter of the pentominoes 12 12 12 10 12 12 © T Madas

The perimeter of the pentominoes 12 12 12 10 12 12 © T Madas

Fitting all the 12 pentominoes in a 6 by 10 rectangle There are 2339

Fitting all the 12 pentominoes in a 6 by 10 rectangle There are 2339 different ways to fit the 12 pentominoes in a 6 by 10 rectangle. Here are 2 more ways: © T Madas

Fitting all the 12 pentominoes in a 5 by 12 rectangle There are 1010

Fitting all the 12 pentominoes in a 5 by 12 rectangle There are 1010 different ways to fit the 12 pentominoes in a 5 by 12 rectangle. Here is another way: © T Madas

Fitting all the 12 pentominoes in a 4 by 15 rectangle There are 368

Fitting all the 12 pentominoes in a 4 by 15 rectangle There are 368 different ways to fit the 12 pentominoes in a 4 by 15 rectangle. Here is another way: © T Madas

© T Madas

© T Madas

© T Madas

© T Madas

Which hexominoes have line symmetry? Which hexominoes have rotational symmetry and of what order?

Which hexominoes have line symmetry? Which hexominoes have rotational symmetry and of what order? Which hexominoes could be the net of a cube? Every hexomino has an area of 6 square units but do they all have the same perimeter? The 35 hexominoes have a total area of 210 square units. By tessellating all 35 hexominoes is it possible to make up: a 14 x 15 rectangle a 10 x 21 rectangle a 7 x 30 rectangle a 6 x 35 rectangle © T Madas

Hexominoes with reflective symmetry © T Madas

Hexominoes with reflective symmetry © T Madas

Hexominoes with rotational symmetry order 2 order 2 © T Madas

Hexominoes with rotational symmetry order 2 order 2 © T Madas

11 hexominoes could be the net of a cube © T Madas

11 hexominoes could be the net of a cube © T Madas

The perimeter of the hexominoes 14 14 12 14 14 14 12 10 14

The perimeter of the hexominoes 14 14 12 14 14 14 12 10 14 14 14 12 14 14 © T Madas

The 35 hexominoes have a total area of 210 units 2. By tessellating all

The 35 hexominoes have a total area of 210 units 2. By tessellating all 35 hexominoes it is NOT possible to make up any rectangle with an area of 210 units 2 © T Madas

How many of each type of Polyomino are there? • Monominoes a 1 •

How many of each type of Polyomino are there? • Monominoes a 1 • Dominoes a 2 • Triominoes a 2 • Tetrominoes a 5 • Pentominoes a 12 • Hexominoes a 35 • Heptominoes a 108 • Octominoes a 369 etc There is no formula which gives the number of all the possible n – ominoes © T Madas

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