Properties are considered independent if none logically implies the other (or its opposite).

Another way of putting it is that on the right, the most descriptive good solution is of the form "[property 1] AND [property 2] vs. neither".

Adjacent-numbered pages:
BP823 BP824 BP825 BP826 BP827  *  BP829 BP830 BP831 BP832 BP833

nice, meta (see left/right), miniproblems, creativeexamples, assumesfamiliarity, structure

boxes_bpimage_three_per_side [smaller | same | bigger]

Aaron David Fairbanks

Left examples have no solution, but they do not break the rules in ways so extreme that it is plainly impossible for them to have a solution, such as including the same image on both sides or including no images per side. (See such as including the same image on both sides or including no images per side.

See BP522 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).

See BP968 (flipped) for a version of this Bongard Problem including examples of invalid Bongard Problems that don't even admit a convoluted solution (the same image appears on both sides).

Also see BP1080, which is similar to BP968, but including various different formats of Bongard Problems, distinguishing them from arbitrary images that are not Bongard Problems.

Adjacent-numbered pages:
BP824 BP825 BP826 BP827 BP828  *  BP830 BP831 BP832 BP833 BP834

nice, meta (see left/right), miniproblems, creativeexamples, left-unknowable, right-narrow, assumesfamiliarity, structure, help, presentationinvariant

boxes_bpimage_three_per_side_nosoln_allowed [smaller | same | bigger]
zoom in right (boxes_bpimage_three_per_side)

Aaron David Fairbanks

For example, in a picture on the left of this Bongard Problem, if object A turned 90 degrees clockwise is object B, then there is also an object C which is B turned 90 degrees clockwise.

Positioning is irrelevant.

In all images, any pair of objects ought to be related in a unique (most intuitive) way. Furthermore, one object is not allowed to be related to two distinct objects by the same relationship. Even for images on the right, each analogy of objects A:B::C:_ should have one clear answer, although that object is perhaps missing.

Relationships described by "[undo-able action] applied to ___ is ___" will always form what in mathematics is called a "group". These relationships can be chained one after another to form a total relationship (turn 90 degrees clockwise + turn 90 degrees clockwise = turn 180 degrees), and each relationship has an "inverse" relationship that undoes it and vice versa (turn 90 degrees clockwise + turn 90 degrees counterclockwise = do nothing).

(Moreover actions are by nature associative.)

Sometimes the relationships in a picture wouldn't be consistently read the same way by everybody. For example, if there is a picture showing an L shape next to all vertical and horizontal reflections and 90 degree rotations of it, somebody might read

⅃ L

to be the same relationship as

┗━

┏━.

Meanwhile, someone else might think ⅃ L should be called the same relationship as ┗━ ━┛. There is a conflict between "flipping over the vertical line within the letter 'L'" and "flipping over a vertical line in the background space."

Likewise in any illustration of related objects (as in this Bongard Problem) people might interpret [the transformation that sends A to B] as analogous to [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].

A "commutative" (also called "abelian") group is a group in which there is no difference between the two in each case. Displayed using pictures like the ones in this Bongard Problem, only commutative groups of relationships can be expected to be read consistently by people.

https://en.wikipedia.org/wiki/Group_(mathematics)

https://en.wikipedia.org/wiki/Abelian_group

See BP842 and BP840 for versions about particular groups.

Adjacent-numbered pages:
BP836 BP837 BP838 BP839 BP840  *  BP842 BP843 BP844 BP845 BP846

nice, rules, miniworlds

zoom in left | zoom in right

Aaron David Fairbanks

Based on Google reCaptcha human verification checks.

Adjacent-numbered pages:
BP839 BP840 BP841 BP842 BP843  *  BP845 BP846 BP847 BP848 BP849

nice, teach, anticomputer, culture, color, experimental, funny

color_street_photograph [smaller | same | bigger]

Cameron Fetter

Or, perhaps more concretely, "Depiction of object with some symmetry (invariance under transformation) vs. depiction of object with no simple symmetries."

Adjacent-numbered pages:
BP842 BP843 BP844 BP845 BP846  *  BP848 BP849 BP850 BP851 BP852

nice, fuzzy, abstract, math, concept, collective, dithering

Leo Crabbe

https://en.wikipedia.org/wiki/Moving_sofa_problem

Adjacent-numbered pages:
BP845 BP846 BP847 BP848 BP849  *  BP851 BP852 BP853 BP854 BP855

nice, precise, physics, creativeexamples, proofsrequired, left-narrow, right-narrow, dithering

Leo Crabbe

Represented are Bongard Problems and an extra object to sort, a.k.a. the Bongard's Dozen format.

A variety of kinds of ambiguities showcased as left examples.

Adjacent-numbered pages:
BP850 BP851 BP852 BP853 BP854  *  BP856 BP857 BP858 BP859 BP860

nice, creativeexamples, structure, rules

bdimage_shapes_extra_below [smaller | same | bigger]
zoom in right (bdimage_shapes_extra_below_clear_sort)

Aaron David Fairbanks

Adjacent-numbered pages:
BP851 BP852 BP853 BP854 BP855  *  BP857 BP858 BP859 BP860 BP861

nice, precise, allsorted, left-narrow, preciseworld

Leo Crabbe

This is a generalisation of BP861.

Adjacent-numbered pages:
BP855 BP856 BP857 BP858 BP859  *  BP861 BP862 BP863 BP864 BP865

hard, nice, stub, precise, stretch, unstable, hardsort, challenge, creativeexamples, perfect, pixelperfect

fill_shape [smaller | same | bigger]

Leo Crabbe

See BP860 for the more general version of this solution.

Adjacent-numbered pages:
BP856 BP857 BP858 BP859 BP860  *  BP862 BP863 BP864 BP865 BP866

nice, precise, unstable, perfect, pixelperfect

fill_shape [smaller | same | bigger]

Leo Crabbe