All examples in this Problem are outlines of polygons or solid black polygons.

M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 215.

BP1211 is "triangle vs. anything else".

Adjacent-numbered pages:
BP1 BP2 BP3 BP4 BP5  *  BP7 BP8 BP9 BP10 BP11

easy, nice, precise, number, ignoreimperfections, finished, traditional, preciseworld, bongard

Multiple options:
polygon_outline_or_fill [smaller | same | bigger],
triangle_or_quadrilateral_outline_or_fill [smaller | same | bigger]

Mikhail M. Bongard

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"). Then, to determine whether an object fits left or right in the Bongard Problem, its value is compared with a fixed threshold value.

Spectra can be continuous or discrete.

A "spectrum" Bongard Problem is usually arbitrary, since there could be made many different versions of it with different choices of threshold value. However, sometimes a certain choice of threshold is particularly natural. For example, the threshold of 90 degrees in "acute vs. obtuse angles" does not come across as arbitrary. And in BP2, the spectrum of values ("size") is vague, so much that the fuzzy threshold, of about half the size of the bounding box, does not seem arbitrary.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of the latter two typically only makes sense when the condition before it is true.

If a spectrum Bongard Problem obeys 1) and 2), then it will usually be precise.

For example:

"Angles less than 90° vs. angles greater than 90°" is "precise".

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will usually be allsorted.

For example:

"Angles less than or equal to 90° vs. angles greater than 90°" is "allsorted".

Discrete spectra usually satisfy 1) but do not satisfy 2). In a discrete spectrum Bongard Problem, there isn't one unambiguous threshold value. Consider "2 or fewer holes vs. 3 or more holes". (Is the threshold 2? 3? 2.5?)

In an especially extreme kind of spectrum Bongard Problem, one side represents just a single value, just the threshold value. For example, "right angles vs. obtuse angles." In certain cases like this the threshold is an extreme value at the very boundary of the spectrum of possible values. For example, consider "no holes vs. one or more holes." Cases like this might not even be understood as two sides of a spectrum, but rather the absence versus presence of a property. (See the keyword notso.)

Even more extreme, in some Bongard Problems, each of the sides is a single value on a spectrum. For example, BP6 is "3 sides vs. 4 sides". We have not been labeling Bongard Problems like this with the keyword "spectrum".

After all, any Bongard Problem can be re-interpreted as a spectrum Bongard Problem, where the spectrum ranges from the extreme fitting left to the extreme of fitting right.

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

See BP874 for the version sorting pictures of Bongard Problems (miniproblems) instead of links to pages on the OEBP.

Adjacent-numbered pages:
BP502 BP503 BP504 BP505 BP506  *  BP508 BP509 BP510 BP511 BP512

notso, meta (see left/right), links, keyword, sideless

bp [smaller | same | bigger]
zoom in left (spectrum_bp)

Aaron David Fairbanks

Bongard Problems sorted left have the keyword "precise" on the OEBP.

Bongard Problems sorted right have the keyword "fuzzy" on the OEBP.

In an precise Bongard Problem, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted.

(All relevant examples clearly sorted either left or right is the keyword allsorted.)

How can it be decided whether or not a rule is precise? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule to determine which examples the original rule intends to sort. Bongard Problems by design communicate ideas without fixing that context ahead of time. The label "precise" can only mean a Bongard Problem's rule seems precise to people who see it. (This "precise vs. fuzzy" Bongard Problem is fuzzy.)

In an precise "less than ___ vs. greater than ___" Bongard Problem (keyword spectrum), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see e.g. BP292).

As a rule of thumb, do not consider imperfections of hand drawn images (keyword ignoreimperfections) when deciding whether a Bongard Problem is precise or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be labelled fuzzy; similar vagueness arises in all hand-drawn Bongard Problems. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword perfect.)

Sometimes the way a Bongard Problem would sort certain examples is an unsolved problem in mathematics. (See e.g. BP820.) There is a precise criterion that has been used to verify each sorted example fits where it fits (some kind of mathematical proof); however, where some examples fit is still unknown. Whether or not such a Bongard Problem should be labelled "precise" might be debated.

(Technical note: some properties are known to be undecidable, and sometimes the decidability itself is unknown. See https://en.wikipedia.org/wiki/Decision_problem .)

(See the keyword proofsrequired.)

One way to resolve this ambiguity is to define "precise" as meaning that once people decide where an example belongs for a reason, they will all agree about it.

Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that, the rule sorting those examples is precise. Say, for some potential new example, it is unclear whether it should be included in the Bongard Problem at all, but, if it were included, it would be clear where it should be sorted (or that it should be left unsorted). A Bongard Problem like this can still be tagged "precise".

(If all examples are clearly sorted except for some example for which it is unclear whether it belongs to the class of relevant examples, the situation becomes ambiguous.)

On the other hand, sometimes the class of all examples is very clear, with an obvious boundary. (Keyword preciseworld.)

There is a subtle distinction to draw between Bongard Problems that are precise to the people making them and Bongard Problems that are precise to the people solving them. A Bongard Problem (particularly a non-allsorted one) might be labeled "precise" on the OEBP because the description and the listed ambiguous examples explicitly forbid sorting certain border cases; however, someone looking at the Bongard Problem without access to the OEBP page containing the definition would not be aware of this. It may or may not be obvious that certain examples were intentionally left out of the Bongard Problem. A larger collection of examples may make it more clear that a particularly blatant potential border case was left out intentionally.

See BP876 for the version with pictures of Bongard Problems instead of links to pages on the OEBP.

See both and neither for specific ways an example can be classified as unsorted in an "precise" Bongard Problem.

Adjacent-numbered pages:
BP503 BP504 BP505 BP506 BP507  *  BP509 BP510 BP511 BP512 BP513

fuzzy, meta (see left/right), links, keyword, right-self, sideless

bp [smaller | same | bigger]

Aaron David Fairbanks

Left-sorted Bongard Problems have the the keyword "left-narrow" on the OEBP.

Call a rule "narrow" if it is likely to be noticed in a large collection of examples, without any counterexamples provided.

A collection of triangles will be recognized as such; "triangles" is a narrow rule. A collection of non-triangular shapes will just be seen as "shapes"; "not triangles" is not narrow.

Intuitively, a narrow rule seems small in comparison to the space of other related possibilities. Narrow rules tend to be phrased positively ("is [property]"), while non-narrow rules opposite narrow rules tend to be phrased negatively ("is not [property]").

Both sides of a BP can be narrow, e.g. BP6.

Even a rule and its conceptual opposite can be narrow, e.g. BP20.

What seems like a typical example depends on expectations. If one is expecting there to be triangles, the absence of triangles will be noticeable. (See the keyword assumesfamiliarity for Bongard Problems that require the solver to go in with special expectations.)

A person might notice the absence of triangles in a collection of just polygons, because a triangle is such a typical example of a polygon. On the other hand, a person will probably not notice the absence of 174-gons in a collection of polygons.

Typically, any example fitting a narrow rule can be changed slightly to no longer fit. (This is not always the case, however. Consider the narrow rule "is approximately a triangle".)

It is possible for a rule to be "narrow" (communicable by a properly chosen collection of examples) but not clearly communicated by a particular collection of examples satisfying it, e.g., a collection of examples that is too small to communicate it.

Note that this is not just BP514 (right-narrow) flipped.

Is it possible for a rule to be such that some collections of examples do bring it to mind, but no collection of examples unambiguously communicates it as the intended rule? Perhaps there is some border case the rule excludes, but it is not clear whether the border case was intentionally left out. The border case's absence would likely become more conspicuous with more examples (assuming the collection of examples naturally brings this border case to mind).

See BP830 for a version with pictures of Bongard Problems (miniproblems) instead of links.

Adjacent-numbered pages:
BP508 BP509 BP510 BP511 BP512  *  BP514 BP515 BP516 BP517 BP518

dual, meta (see left/right), links, keyword, side

bp [smaller | same | bigger]

Aaron David Fairbanks

See BP6 for "triangle vs. quadrilateral".

Adjacent-numbered pages:
BP1206 BP1207 BP1208 BP1209 BP1210  *  BP1212 BP1213 BP1214 BP1215 BP1216

stub, notso, left-narrow, traditional

Aaron David Fairbanks