OU on the BBC: The Code - Master of Mosaics hints and tips

1. Master of Mosaics is about the symmetries of patterns known as wallpaper patterns. Wallpaper patterns are characterised by repetitive design.
2. There are four different types of symmetries. Master of Mosaics is about rotational symmetries. A rotational symmetry of a mosaic pattern is a rotation that leaves the pattern unchanged. The other types of symmetries, which you do not need to identify in Master of Mosaics, are translations, reflections, and glide reflections.
3. There are two components to a rotation, which are a centre of rotation and an angle of rotation. You have control of both of these in Master of Mosaics.
4. Choose a point that looks like a centre of rotation and click and hold the mouse on that point. A circle appears with an angle labelled, which changes as you move the mouse around. Take the mouse away from the centre of rotation so that you have more control. Slowly work your way from 0°  to 360° looking for angles of rotation.
5. The only possible angles of rotation are 60° , 90° , 120° , 180° , 240° , 270°, and 300°, so hover around these numbers. (These numbers are multiples of 60, 90, 120, and 180.)
6. If you find a rotational symmetry of angle 60°  at a point then all the rotational symmetries at that point are 60° , 120° , 180°, 240°, and 300°  (multiples of 60) . If you find a rotational symmetry of angle 90°  then all the rotational symmetries at that point are 90° , 180° , and 270°  (multiples of 90). The other two possibilities are that there are two rotational symmetries of angles 120° and 240°  (multiples of 120) or just a single rotational symmetry of angle 180° .
7. Once you have found all the rotational symmetries at a particular point, move on to another point. You cannot move on to another point which looks similar to the point you are already on, because the rotational symmetries at similar points are essentially the same.
8. The collection of all symmetries of a wallpaper pattern is called a wallpaper group. A group is an important mathematical object. It is a remarkable fact that there are essentially only 17 wallpaper groups. Every wallpaper pattern has a corresponding wallpaper group, and this wallpaper group is one of 17 possible types.
9. The seventeen wallpaper groups are called p1, p2, pm, pg, cm, pmm, pmg, pgg, cmm, p4, p4m, p4g, p3, p3m1, p31m, p6, and p6m. These names describe features of the groups. In Master of Mosaics, the name appears in the top right hand corner of the screen. Below we list the number of different types of rotational symmetries in each group. We write '90° symmetry' to refer to all multiples of 90° (that is, 90, 180, and 270).

p1 : no rotational symmetries
p2 : four 180° symmetries
pm : no rotational symmetries
pg : no rotational symmetries
cm : no rotational symmetries
pmm : four 180° symmetries
pmg : two 180° symmetries
pgg : two 180° symmetries
cmm : three 180° symmetries
p4 : two 90° symmetries, one 180° symmetry
p4m : two 90° symmetries, one 180° symmetry
p4g : two 90° symmetries, two 180° symmetries
p3 : three 120° symmetries
p3m1 : three 120° symmetries
p31m : three 120° symmetries
p6 : one 60° symmetry, two 120° symmetries, three 180 degree° symmetries
p6m : one 60° symmetry, two 120° symmetries, three 180° symmetries

10. Here is a list of the wallpaper groups corresponding to each level in Master of Mosaics.

Level 1 : pgg
Level 2 : p4g
Level 3 : pmm
Level 4 : pmm
Level 5 : p4m
Level 6 : p31m
Level 7 : cmm
Level 8 : p3
Level 9 : p6m
Level 10 : p6
Level 11 : p4
Level 12 : p6
Level 13 : p3m1
Level 14 : p6
Level 15 : pmg
Level 16 : p2

In Level 2 you only have to find one 90° symmetry and one 180° symmetry, despite the group being p4g. This is because the two 90° symmetries look similar, and so do the two 180° symmetries.

In Level 6 you only have to find two of the three 120° symmetries (two of them look similar).

In Levels 9, 10, 12, and 14 you only have to find one 60° symmetry, one 120° symmetry, and one 180° symmetry. (The two 120° symmetries look similar, as do the three 180° symmetries.)

Want a harder version of Master of Mosaics? Why not try the Open University's Grandmaster of Mosaics game!