3.9 Spatial reasoning 113
over and over again, filling any area without
any gaps, to give the pattern shown in the
original drawing. So two white tiles are needed
for every black tile. B is the correct answer.
The activity below involves three-
dimensional reasoning. Because the drawing is
not of a familiar object, there are no short-
cuts; you need to work out what the
possibilities are for the unseen side.
Commentary
There must be some sort of recess in the
top-left corner (top back right as shown on the
3-D drawing). We cannot tell whether it goes
right to the bottom as shown in A and B or just
some of the way down as shown in C, D and E.
Similarly, there is a recess that goes through
to the right-hand edge (left back on the 3-D
drawing). Those shown in A, C and E would all
give the same 3-D view from the front. We can
now eliminate A, C and E, as both the rear
features are shown as being possible.
(Remember that we are looking for the
diagram that is not a possible representation.)
D joins up the two recesses – this would also be
possible as the join would not show from the
angle originally shown.
We have come to the answer B by
elimination. This is a completely valid way of
proceeding, but it would be useful to check
that this is indeed the correct answer. The
recess shown in green on the right-hand side
of diagram B would be visible at the back of
the lowest section of the three-dimensional
drawing. So B is not possible.
Once again, this is more difficult than
might be expected. We do not actually know
what the hidden reverse side looks like – there
are an infinite number of possibilities. All we
can do is consider what hints are given by the
three-dimensional picture. One primary
feature of this type of question is that the
answer cannot be produced just by
considering the information given and the
question. This is a backward question. The five
options must all be looked at and a decision
made on whether each is possible. Backward
questions are a regular feature of questions on
spatial reasoning, and of identifying
similarity, which was dealt with in Chapter 3.7.
They do not occur very often in the other
types of question. The value of elimination
was shown in the method of answering this
type of question.The drawing is a three-dimensional
representation of a puzzle piece.Which diagram is not a possible
representation of how it looks from the back
(i.e. the direction shown by the arrow)? The
shaded areas are recesses.A BCDEActivity