M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 229.

Adjacent-numbered pages:
BP41 BP42 BP43 BP44 BP45  *  BP47 BP48 BP49 BP50 BP51

finished, traditional, viceversa, bongard

Mikhail M. Bongard

This is the final problem in Bongard's original collection. It is the only member of the collection that makes reference to human culture. This can be interpreted symbolically as foreshadowing that computers will be able to perform the various tasks that humans can do.

Another idea introduced by this Bongard Problem is that a Bongard Problem can teach its solution to the solver. (See keyword teach.) A large pool of examples can be used for training, as is common in machine learning.

M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 247.

Adjacent-numbered pages:
BP95 BP96 BP97 BP98 BP99  *  BP101 BP102 BP103 BP104 BP105

easy, nice, teach, arbitrary, anticomputer, culture, finished, bongard

Mikhail M. Bongard

All examples are connected smooth curves allowed to self-intersect that must curve in only one direction (starting at one end, either clockwise or counter-clockwise), i.e. there is no inflection point.

Adjacent-numbered pages:
BP208 BP209 BP210 BP211 BP212  *  BP214 BP215 BP216 BP217 BP218

less, convoluted, perfect, traditional

Giuseppe Insana

Examples on the left are also known as "Dyck words".

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

Adjacent-numbered pages:
BP951 BP952 BP953 BP954 BP955  *  BP957 BP958 BP959 BP960 BP961

easy, nice, precise, allsorted, unwordable, notso, sequence, traditional, inductivedefinition, preciseworld, left-listable, right-listable

Aaron David Fairbanks

Left-sorted examples divide the plane into a maximal amount of disconnected white regions by a given number of "cuts". The number of regions in one of these examples will be the nth value of of the Lazy Caterer sequence ( https://oeis.org/A000124 ), where n is the number of lines.

https://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence

Adjacent-numbered pages:
BP1262 BP1263 BP1264 BP1265 BP1266  *  BP1268 BP1269 BP1270 BP1271 BP1272

precise, allsorted, notso, perfect

Leo Crabbe

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

Adjacent-numbered pages:
BP1170 BP1171 BP1172 BP1173 BP1174  *  BP1176 BP1177 BP1178 BP1179 BP1180

precise, traditional, grid, miniworlds, dithering

Leo Crabbe

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

This is a generalisation of Bongard Problems that allows them to have any number of sides. There is a sense in which this problem is about valid vs. invalid ways of partitioning a set of examples into equivalence classes.

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

Adjacent-numbered pages:
BP1148 BP1149 BP1150 BP1151 BP1152  *  BP1154 BP1155 BP1156 BP1157 BP1158

abstract, teach, meta (see left/right), miniproblems, infodense, structure, rules, miniworlds

zoom in left

Leo Crabbe

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

Adjacent-numbered pages:
BP535 BP536 BP537 BP538 BP539  *  BP541 BP542 BP543 BP544 BP545

Jago Collins

https://en.wikipedia.org/wiki/Hex_(board_game)

Adjacent-numbered pages:
BP1240 BP1241 BP1242 BP1243 BP1244  *  BP1246 BP1247 BP1248 BP1249 BP1250

hard, precise, convoluted, dual, rotate, boundingbox, hardsort, challenge, proofsrequired, bordercontent

Aaron David Fairbanks