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

Each example in this Bongard Problem consists of mini-panels containing the same arrangement of circles (ignoring colouring). Each mini-panel has a single circle highlighted in red, and possibly some circles highlighted in blue. A strict rule for this Bongard Problem could be something like "If a circle is blue in one mini-panel and red in a second mini-panel, then there are no blue circles in the second mini-panel that weren't already blue in the first mini-panel." The relation interpretation is that a circle is related to the red circle if and only if it is coloured blue.

Adjacent-numbered pages:
BP968 BP969 BP970 BP971 BP972  *  BP974 BP975 BP976 BP977 BP978

convoluted, color, infodense, rules

Jago Collins

Each example in this Bongard Problem consists of mini-panels containing the same arrangement of circles (ignoring colouring). Each mini-panel has a single circle highlighted in red, and possibly some circles highlighted in blue. A strict rule for this Bongard Problem could be something like "If a circle is blue in one mini-panel and red in a second mini-panel, then the red circle from the first mini-panel is blue in the second mini-panel." The relation intepretation is that a circle is related to the red circle if and only if it is coloured blue. BP973 is a similar problem.

Adjacent-numbered pages:
BP970 BP971 BP972 BP973 BP974  *  BP976 BP977 BP978 BP979 BP980

convoluted, color, infodense, rules

Jago Collins

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

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually becoming ___"; gradually becoming thickly outlined.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually becoming "gradually becoming ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and call it "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more belabored way of putting it:

Call something like "is star-shaped" a "rule". An object can fit a rule.

Call something like "all but one are ___" a "rule-parametrized rule". A collection of objects, with respect to a particular rule, can fit a rule-parametrized rule.

A drawing on the right shows many collections. Every collection fits the same rule-parametrized rule (with respect to various rules); furthermore the collection of collections fits that same rule-parametrized rule (with respect to some rule collections can fit).

Likewise a drawing on the left shows a collection of collections, with some noticeable recurring rule-parametrized rule. The collection of collections must fit that rule-parametrized rule with respect to the rule of fitting that rule-parametrized rule (with respect to various rule).

An unintended solution to this BP is "not all groups share some noticeable property vs. all do." It is hard to come up with examples foiling this alternative solution because (what was above called) the rule-parametrized rule usually has to do with not all objects in the collection fitting the rule. (See BP568, which is about BP ideas that are always overridden by a simpler solution.)

Some examples would fit left under a certain interpretation: EX8220 "all are 'all are ___' " and EX8222 "palindrome with respect to being a palindrome with respect to ___" (every shown collection is a palindrome with respect to some property, and all things in a list being the same is a palindrome). But those rules are not necessarily the most obvious ways of interpreting these pictures, so they have been marked as ambiguous. Either of these placed on the left would prevent the intended solution being overridden (see the previous paragraph).

Here is a list of left example ideas that would be impossible to make:

- Exhaustive list of all exhaustive lists of all ____.

The right side of this Problem is a subset of BP999^left.

Adjacent-numbered pages:
BP993 BP994 BP995 BP996 BP997  *  BP999 BP1000 BP1001 BP1002 BP1003

"Odd one out with respect to what property is the odd one out" would not fit left in this Problem: even though this example does seem doubly-meta, it is not doubly-meta in the right way. There is no odd one out with respect to the property of having an odd one out.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the __ appearing in 'X __'] ". Examples like these two could make for a different Bongard Problem.

hard, unwordable, challenge, overriddensolution, infodense, contributepairs, funny, rules, miniworlds

zoom in right

Aaron David Fairbanks

Rhetorical question: Where would the collection of left examples of this Bongard Problem be sorted by this Bongard Problem? (The question is whether these examples considered together satisfy the pattern that all the parts do, namely that the whole satisfies the pattern that all the parts do.)

See BP793 and BP1004 for similar paradoxes.

See BP1005 for the version about only numerical properties; examples in that BP would be sorted the same way here that they are there.

See BP1003 for a similar idea. Rather than the collection of collections imitating the individual collections, BP1003 is about the total combined collection imitating the individual collections. A picture showing (for example) an odd number of even-numbered groups would be sorted differently by these two BPs.

Also see BP1004, is likewise about the whole satisfying the same rule as its parts, but there the parts don't themselves have to be collections; there the parts are just plain individual objects. The panels in BP999 (this BP) should be sorted the same way in BP1004.

See BP1002, which is about only visual self-similarity instead of more general conceptual "self-similarity".

Adjacent-numbered pages:
BP994 BP995 BP996 BP997 BP998  *  BP1000 BP1001 BP1002 BP1003 BP1004

nice, stub, abstract, creativeexamples, left-narrow, rules, miniworlds

[smaller | same | bigger]
zoom in left | zoom in right

Aaron David Fairbanks

Since it is most intuitive to imagine spatially squishing together all the collections in the process of combining them into one big collection, avoid rules that involve relative spatial positionings of objects.

Contrast BP999, which is very similar. There, when considering the whole picture, the collections are to be treated as individual objects; here, when considering the whole picture, the collections are to be combined into one big collection. A picture showing (for example) an odd number of even-numbered groups would be sorted differently by these two BPs.

Also contrast BP1004, which is about a collection of plain objects obeying the same rule as all the objects (instead of a collection of [collections of objects] obeying the same rule as all the [collections of objects]).

See BP1006 for the version with only number-based properties. All panels in that Bongard Problem fit the same way in this Bongard Problem as well.

Adjacent-numbered pages:
BP998 BP999 BP1000 BP1001 BP1002  *  BP1004 BP1005 BP1006 BP1007 BP1008

nice, abstract, notso, creativeexamples, rules, miniworlds

[smaller | same | bigger]

Leo Crabbe, Aaron David Fairbanks

The "whole" is the entire panel including the bounding box. A "part" is some region either stylistically different or amply separated in space from everything else. Smaller parts-within-parts don't count as parts.

Rhetorical question: Where would the collection of left examples of this Bongard Problem be sorted by this Bongard Problem? (The question is whether these examples considered together satisfy the pattern that all the parts do, namely that the whole satisfies the pattern that all the parts do.)

See BP793 and BP999 for similar paradoxes.

See BP1006 for the version about numerical properties where each part is a cluster of dots; examples in that BP would be sorted the same way here that they are there.

See BP999 and BP1003 for versions where each object is itself a collection of objects, so that the focus is on rules specifically pertaining to collections (e.g. "all the objects are different").

See BP1002 for a Bongard Problem about only visual self-similarity instead of conceptual self-similarity.

The rule shown in each panel is "narrow" (see BP513^left and BP514^left).

Adjacent-numbered pages:
BP999 BP1000 BP1001 BP1002 BP1003  *  BP1005 BP1006 BP1007 BP1008 BP1009

nice, abstract, anticomputer, creativeexamples, left-narrow, rules, miniworlds

Aaron David Fairbanks