Identifying strings from different viewpoints
(Difficult but not impossible!)

This is part of a collection of examples of abilities of humans to reason in a
mathematical way even if they don't know they are doing it. See the 'knots' page.

Each picture shows a scene containing a continuous string looped over and
around itself in various ways.

Each scene was photographed from two or more viewpoints. The picture sequence
has been scrambled, so that adjacent pictures are not always pictures of the
same scene.

Can you group together the pictures that were of the same scene? (Please ignore
the knot at the visible free end of the string, required for a different use of the

Some of the pairs are quite difficult to identify. You need to infer some of the
3-D structure instead of using only the 2-D image structure.

Could a robot be given the ability to tell which pictures are of the same
scene? How?

Would training on many pairs of images suffice, or would the robot have to be
given explicitly a comparison procedure? Or could it discover such a procedure
itself? Could the procedure be expressed in a logical language, e.g. predicate

If not what alternatives are possible?

NOTE 1: if the string in a particular configuration is construed as a "Possibility
transducer". the inputs to the transducer would be selection of viewpoints and
direction of gaze and the outputs could be a specification of the relationship
to the previous scene, if the objects are recognized as the same, otherwise a
simple 'No'.

NOTE 2: in what way does deciding whether two images are photographs of the same
configuration require mathematical reasoning?

Can you group these into pictures of the same 3-D scene taken from different

    String A:
    Scene 2a
    String B:
    Scene 3a
    String C:
    Scene 1
    String D:
    Scene 4a
    String E:
    Scene 6a
    String F:
    Scene 5a
    String G:
    Scene 2b
    String H:
    Scene 3b
    String I:
    Scene 4b
    String J:
    Scene 5b
    String K:
    Scene 6b
    String L:
    Scene 5c
    String M:
    Scene 6c
    String N:
    Scene 4c
For more examples, see the 'Toddler Theorems' Web page: