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

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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

Adjacent-numbered pages:
BP1044 BP1045 BP1046 BP1047 BP1048  *  BP1050 BP1051 BP1052 BP1053 BP1054

teach, creativeexamples, left-narrow, right-narrow, contributepairs, fixedgrid, miniworlds

Jago Collins

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

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

This is a special case of BP841 and a generalisation of BP822.

Adjacent-numbered pages:
BP1105 BP1106 BP1107 BP1108 BP1109  *  BP1111 BP1112 BP1113 BP1114 BP1115

nice, abstract, math, anticomputer, creativeexamples, left-narrow, unorderedpair, rules, miniworlds, dithering

Leo Crabbe

All examples in this Problem are grids consisting of two objects.

Adjacent-numbered pages:
BP1118 BP1119 BP1120 BP1121 BP1122  *  BP1124 BP1125 BP1126 BP1127 BP1128

EX9124 shows a 9 square by 9 square grid. Take each tile to be 3 squares by 3 squares; there is a 3 tile by 3 tile checkerboard pattern. (One of these tiles is itself a checkerboard pattern; the other is all black squares.)

hard, nice, precise, allsorted, hardsort, grid, miniworlds

Jago Collins

Other ways of phrasing this:

"Local" vs. "global" properties of collections: to check a collection satisfies a "local" property, it is only necessary to check each individual thing in it satisfies some property.

The rule all collections satisfy is just "every object is a ___" vs. the rule is something more.

Adjacent-numbered pages:
BP1122 BP1123 BP1124 BP1125 BP1126  *  BP1128 BP1129 BP1130 BP1131 BP1132

abstract, creativeexamples, left-unknowable, rules, miniworlds

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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, infodense, structure, rules, miniworlds

zoom in left

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