Adjacent-numbered pages:
BP360 BP361 BP362 BP363 BP364  *  BP366 BP367 BP368 BP369 BP370

traditional, rules, miniworlds

constant_change_seq_increase_right [smaller | same | bigger]

Aaron David Fairbanks

Adjacent-numbered pages:
BP368 BP369 BP370 BP371 BP372  *  BP374 BP375 BP376 BP377 BP378

abstract, anticomputer, concept, creativeexamples, left-narrow, right-narrow, contributepairs, traditional, miniworlds, dithering

Aaron David Fairbanks

Related to BP380 and BP792.

Adjacent-numbered pages:
BP374 BP375 BP376 BP377 BP378  *  BP380 BP381 BP382 BP383 BP384

nice, abstract, traditional, rules, miniworlds

collection_of_objects_same_type [smaller | same | bigger]

Aaron David Fairbanks

"True" vs. "false."

Adjacent-numbered pages:
BP388 BP389 BP390 BP391 BP392  *  BP394 BP395 BP396 BP397 BP398

nice, fuzzy, abstract, collective, contributepairs, traditional, rules, miniworlds, dithering

Jago Collins

Related to BP379 and BP380.

EX6572 can ambiguously be taken to be self-referential as the set of left-hand examples (which would make the right hand solution only "incomplete sets") or the class of finite sets.

Adjacent-numbered pages:
BP787 BP788 BP789 BP790 BP791  *  BP793 BP794 BP795 BP796 BP797

nice, abstract, creativeexamples, rules, miniworlds

collection_of_objects_same_type [smaller | same | bigger]

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 "[undoable 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 compound 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

All examples depict a process that transforms one object into another (two example input-output pairs are provided in every panel). In left-sorted examples, each input corresponds to a unique output, whereas in right-sorted examples, different inputs could potentially lead to the same output. There is a sense in which all the processes described on the right "lose" some amount of the input's information.

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

Adjacent-numbered pages:
BP912 BP913 BP914 BP915 BP916  *  BP918 BP919 BP920 BP921 BP922

nice, abstract, creativeexamples, structure, rules, miniworlds

Leo Crabbe

There are many ambiguities here. The solver is expected to determine what things are "allowed" to be inputs for each process. To avoid confusion examples should not be sorted differently if you consider inputting nothing.

In each example there is at least some overlap between the set of possible inputs and the set of possible outputs for each process. If we did not apply this constraint, an easy example to be sorted right would be a process that turns blue shapes red.

A harder-to-read but more clearly defined version of this Problem could include within each example a mini Bongard Problem sorting left all allowed inputs for the process.

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

Adjacent-numbered pages:
BP946 BP947 BP948 BP949 BP950  *  BP952 BP953 BP954 BP955 BP956

structure, rules, miniworlds

Leo Crabbe

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole. One square from somewhere along the edge of the grid is removed.

Intentionally left out of this Problem (shown above sorted ambiguously) are cases in which the rule is not possible to deduce without seeing more squares. Due to this choice to omit those kinds of examples from the right, another acceptable solution to this Problem is "it is possible to deduce the contents of the missing square once the underlying rule is understood vs. not so."

Adjacent-numbered pages:
BP974 BP975 BP976 BP977 BP978  *  BP980 BP981 BP982 BP983 BP984

structure, rules, miniworlds

grid_of_images_with_rule_one_on_edge_missing [smaller | same | bigger]

Aaron David Fairbanks

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

To word the solution with mathematical jargon, "defines a (simply described) map from the Cartesian product of two sets vs. not so." Another equivalent solution is "columns (alternatively, rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy grids seen in BP361. Any analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" might be about how the images must relate to their neighbors, for example.

There is a trivial way in which any example can be interpreted so that it fits on the left side: imagine each row is assigned the list of all the squares in that row and each column is assigned its number, counting from the left. But each grid has a clear rule that is simpler than this.

See BP979 for use of similar structures but with one square removed from the grid. Examples on the left here with any square removed should fit on the left there.

Adjacent-numbered pages:
BP976 BP977 BP978 BP979 BP980  *  BP982 BP983 BP984 BP985 BP986

stub, convoluted, teach, structure, rules, grid, miniworlds

grid_of_images_with_rule [smaller | same | bigger]
zoom in left (grid_of_operations)

Aaron David Fairbanks