All examples are black and white images, partitioned by lines such that crossing a line switches the background color and the foreground color. (Sometimes it is not clear which is "background" and which is "foreground".) In the space between two dividing lines, there is a black and white scene; the outlines of the shapes are curves dividing black from white. Images sorted left are such that each outline-curve present in a scene that comes in contact non-tangentially with a dividing line continues across the dividing line, across which the black and white sides of it switch.

Examples (especially right) usually have ambiguity to some degree; depending on how a person reads the images, dividing lines may be confused for curves within a scene.

Adjacent-numbered pages:
BP519 BP520 BP521 BP522 BP523  *  BP525 BP526 BP527 BP528 BP529

fuzzy, unwordable, anticomputer, traditional, blackwhiteinvariant

Aaron David Fairbanks

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

Right-sorted BPs have the keyword "gap" on the OEBP.

A Bongard Problem with a gap showcases two completely separate classes of objects.

For example, the Bongard Problem "white vs. black" (BP962) has a gap; there is no obvious choice of shared context between the two sides. One could imagine there is a spectrum of grays between them, or that there is a space of partially filled black-and-white images between them, or any number of other ambient contexts.

Bongard Problems about comparing quantities on a spectrum should not usually be considered "gap" BPs. (Discrete spectra perhaps.) A spectrum establishes a shared context, with examples on both sides of the BP landing somewhere on it. (However, if it is reasonable to imagine getting the solution without noticing a spectrum in between, it could be a gap, since the ambient context is unclear.)

Bongard Problems with gaps may seem particularly arbitrary when the two classes of objects are particularly unrelated.

If a Bongard Problem has a "gap" it is likely precise: it will likely be clear on which side any potential example fits.

"Gap" implies stable. (This technically includes cases in which ALL small changes make certain examples no longer fit in with the Bongard Problem, as is sometimes the case in "gap" BPs. See also BP1144.)

See also preciseworld. "Gap" Bongard Problems would be tagged "preciseworld" when the two classes of objects are each clear; it is then apparent that there is no larger shared context and that no other types of objects besides the two types would be sorted by the Bongard Problem.

See BP1140, which is about any (perhaps large) additions instead of repeated small changes.

Adjacent-numbered pages:
BP959 BP960 BP961 BP962 BP963  *  BP965 BP966 BP967 BP968 BP969

unwordable, meta (see left/right), links, keyword, sideless, invariance

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

The description in terms of rectangles was noted by Sridhar Ramesh when he solved this.

All examples in this Bongard Problem feature arced line segments connected at endpoints; these segments do not cross across one another and they are nowhere vertical; they never double back over themselves in the horizontal direction.

Furthermore, in each example, there is a single leftmost point and a single rightmost point, and every segment is part of a chain bridging between them. So, there is a topmost total chain of segments and bottommost total chain of segments.

Any picture on the left can be turned into a subdivided rectangle by the process of expanding points into vertical lines.

Here is another answer:

"Right examples: some junction point has a single line coming out from either the left or right side."

If there is some junction point with only a single line coming out from a particular side, the point cannot be expanded into a vertical segment with two horizontal segments bookending its top and bottom (as it would be if this were a subdivision of a rectangle).

And this was the original, more convoluted idea of the author:

"Start with a string along the top path. Sweep it down, region-by-region, until it lies along the bottom path. The string may only enter a region when it fully covers that region's top edge and likewise it must exit by fully covering the bottom edge. Only in left images can this process be done so that no segment of the string ever hesitates."

Quite convoluted when spelled out in detail. (Although it is not terribly complicated to imagine visually. See the keyword unwordable.)

The string-sweeping answer is the same as the rectangle answer because a rectangle represents the animation of a string throughout an interval of time. (A horizontal cross-section of the rectangle represents the string, and the vertical position is time.) Distorting the rectangle into a new shape is the same as animating a string sweeping across that new shape.

In particular, shrinking vertical lines of a rectangle into points means just those points of the string stay still as the string sweeps down.

The principle that horizontal lines subdividing the original rectangle become the segments in the final picture corresponds to the idea that the string must enter or exit a single region all at once.

BP1129 started as an incorrect solution for this Bongard Problem. Anything fitting right in BP1130 fits right in BP1129.

Adjacent-numbered pages:
BP1125 BP1126 BP1127 BP1128 BP1129  *  BP1131 BP1132 BP1133 BP1134 BP1135

hard, unwordable, solved

[smaller | same | bigger]

Aaron David Fairbanks

Left-sorted problems have the keyword "finishedexamples" on the OEBP.

The addition does not have to be slight.

Left-sorted Problems usually have a very specific collection of examples, where the only images sorted all show the same type of object.

Any Bongard Problem where all examples are one shape outline will be sorted left, and (almost) any Bongard Problem where all examples are one fill shape will be sorted right.

See BP1144 for the version about both additions and erasures, and only slight changes are considered.

See BP1167 for a stricter version, the condition that all examples have the same amount of black and white.

Adjacent-numbered pages:
BP1137 BP1138 BP1139 BP1140 BP1141  *  BP1143 BP1144 BP1145 BP1146 BP1147

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

Leo Crabbe

Left-sorted examples are sometimes called self-descriptive sequences.

See BP1147 for a similar idea.

BP1149 was inspired by this.

Adjacent-numbered pages:
BP1143 BP1144 BP1145 BP1146 BP1147  *  BP1149 BP1150 BP1151 BP1152 BP1153

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

Leo Crabbe

Inspired by BP1148.

Adjacent-numbered pages:
BP1144 BP1145 BP1146 BP1147 BP1148  *  BP1150 BP1151 BP1152 BP1153 BP1154

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

Aaron David Fairbanks

Adjacent-numbered pages:
BP1150 BP1151 BP1152 BP1153 BP1154  *  BP1156 BP1157 BP1158 BP1159 BP1160

abstract, unwordable, creativeexamples, right-unknowable, traditional, finishedexamples, rules

Leo Crabbe

Operations depicted in right-sorted examples are called "commutative".

"Order matters" here means that if the objects in the top half were to switch places, the output would look different.

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

Adjacent-numbered pages:
BP1152 BP1153 BP1154 BP1155 BP1156  *  BP1158 BP1159 BP1160 BP1161 BP1162

nice, abstract, unwordable, notso, structure, rules, miniworlds

Leo Crabbe