Thinking Skills: Critical Thinking and Problem Solving

5.3 Carrying out investigations 221

1 23

45 = 2

6 = 6 7

= 58= 3

= 7

There are eight distinguishably different tiles

in total.

One symmetrical 4 × 4 pattern is shown

below. There are many others. Remember that

the colours must match at the edges.

As an additional activity, you might look at all

similar tiles which join two one-third-edges

instead of half-edges, an example of which is

shown below. There are not so many of these

to make the problem too long, but it will take a

little more care to identify all the different

ones. To start with, how many options do you

need to look at?

Commentary
It is possible to count the maximum number of
tiles that there could be. Half-edge 1 could be
connected to itself or any of the other seven
half-edges. (Note that the blank tile is equivalent
to connecting a half-edge to itself or the adjacent
half-edge on the same side of the square.)
Half-edge 2 could be connected to any other
than half-edge 1 and itself (both of which we
have already counted). Thus, we must
investigate 7 + 6 possibilities. Beyond this,
looking at connections from half-edges 3, 4 etc.
all will produce rotations of those already found.
The full set of 13 possibilities is illustrated
below. The numbers below the tiles give a
successive count of new ones. Where it says,
for example, ‘= 2’, this means that the tile is
equivalent (i.e. can be rotated into) tile 2.