On the left, each row and column could be labeled by a certain object or concept; on the right this is not so.

More specifically: on the left, each row and each column is associated with a certain object or concept; there is a rule for combining rows and columns to give images; it would be possible without changing the rule to extend with new rows/columns or delete/reorder any existing columns. On the right, this is not so; the rule might be about how the images must relate to their neighbors, for example.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey.

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 here.

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.

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.

BP1258 is a similar idea: "any square removed could be reconstructed vs. not." Examples included left here usually fit left there, but some do not e.g. EX9998.

See BP979 for use of similar structures but with one square removed from the grid.

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

nice, convoluted, unwordable, notso, teach, structure, rules, grid, miniworlds

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

Aaron David Fairbanks

All examples in this Problem are sequences of graphic symbols. In this Problem, a "palindrome" is taken to be an ordered sequence which is the same read left-to-right as it is read right-to-left. A more formal solution to this Problem could be: "Sequences which are invariant under a permutation which swaps first and last entries, second and second last entries, third and third last entries, ... and so on vs. sequences which are not invariant under the aforementioned permutamation."

Adjacent-numbered pages:
BP981 BP982 BP983 BP984 BP985  *  BP987 BP988 BP989 BP990 BP991

nice, precise, allsorted, notso, sequence, traditional

zoom in left | zoom in right

Jago Collins

All examples in this Problem are solid concave black shapes. In this Problem, the "cavities" of a concave shape are defined to be the convex hull of the shape minus the shape itself. For example, if you take a bite out of the edge of a piece of paper, the piece of paper in your mouth is the cavity of the bitten piece of paper. The idea may be indefinitely extended, considering whether the cavities of the cavities are concave or convex, and so on.

Adjacent-numbered pages:
BP987 BP988 BP989 BP990 BP991  *  BP993 BP994 BP995 BP996 BP997

nice, precise, perfect, traditional

concave_fill_shape [smaller | same | bigger]

Jago Collins

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, which 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, abstract, creativeexamples, left-narrow, rules, miniworlds

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Aaron David Fairbanks

See BP1004 for a Problem about conceptual self-similarity instead of visual self-similarity.

See BP188 for a similar Problem restricted to shape outlines made of shape outlines.

Adjacent-numbered pages:
BP997 BP998 BP999 BP1000 BP1001  *  BP1003 BP1004 BP1005 BP1006 BP1007

easy, nice, fuzzy, abstract, anticomputer, concept, traditional

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

This is a version of BP999 with only numbers.

Contrast BP1006, which is very similar.

Adjacent-numbered pages:
BP1000 BP1001 BP1002 BP1003 BP1004  *  BP1006 BP1007 BP1008 BP1009 BP1010

nice, notso, hardsort, left-narrow, rules

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Aaron David Fairbanks

This is a version of BP1003 with only numbers.

Contrast BP1005, which is very similar.

Adjacent-numbered pages:
BP1001 BP1002 BP1003 BP1004 BP1005  *  BP1007 BP1008 BP1009 BP1010 BP1011

nice, hardsort, left-narrow, rules

[smaller | same | bigger]
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Leo Crabbe, Aaron David Fairbanks

Henneberg, L. (1911), Die graphische Statik der starren Systeme, Leipzig

Jackson, Bill. (2007). Notes on the Rigidity of Graphs.

Laman, Gerard. (1970), "On graphs and the rigidity of plane skeletal structures", J. Engineering Mathematics, 4 (4): 331–340.

Pollaczek‐Geiringer, Hilda (1927), "Über die Gliederung ebener Fachwerke", Zeitschrift für Angewandte Mathematik und Mechanik, 7 (1): 58–72.

Adjacent-numbered pages:
BP1011 BP1012 BP1013 BP1014 BP1015  *  BP1017 BP1018 BP1019 BP1020 BP1021

nice, physics, help

planar_connected_graph [smaller | same | bigger]
zoom in left (rigid_planar_connected_graph)

Aaron David Fairbanks