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BP1209 Empty square present vs. no square present.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Nearly identical to BP1221, which is the same except non-empty squares are allowed.

See BP24 for "circle present vs. not".

Adjacent-numbered pages:
BP1204 BP1205 BP1206 BP1207 BP1208  *  BP1210 BP1211 BP1212 BP1213 BP1214

KEYWORD

stub, left-narrow, traditional

CONCEPT empty (info | search),
existence (info | search),
square (info | search)

AUTHOR

Aaron David Fairbanks

BP1208 More triangles left than right vs. more triangles right than left.
(edit; present; nest [left/right]; search; history)
COMMENTS

The triangles vary in size in order to make the solution clearly about quantity, not "total mass".

CROSSREFS

Adjacent-numbered pages:
BP1203 BP1204 BP1205 BP1206 BP1207  *  BP1209 BP1210 BP1211 BP1212 BP1213

KEYWORD

stub, precise, spectrum, dual, handed, leftright, traditional

CONCEPT number (info | search),
triangle (info | search),
quantity_comparison (info | search)

AUTHOR

Aaron David Fairbanks

BP1207 Horizontal axis of symmetry vs. no horizontal axis of symmetry.
(edit; present; nest [left/right]; search; history)
CROSSREFS

BP1206 is the same solution but using the vertical axis instead of the horizontal axis.

BP1215 is the same solution but with the NW/SE diagonal instead of the vertical axis.

Adjacent-numbered pages:
BP1202 BP1203 BP1204 BP1205 BP1206  *  BP1208 BP1209 BP1210 BP1211 BP1212

KEYWORD

stub, notso, stretch, left-narrow, traditional

CONCEPT horizontal (info | search),
symmetry_axis (info | search),
symmetry (info | search)

AUTHOR

Aaron David Fairbanks

BP1206 Vertical axis of symmetry vs. no vertical axis of symmetry.
(edit; present; nest [left/right]; search; history)
COMMENTS

The solution for this Bongard Problem is also a (less specific) solution for BP500, "vertical axis of symmetry vs. no axis of symmetry".

CROSSREFS

BP1207 is the same solution but using the horizontal axis instead of the vertical axis.

BP1215 is the same solution but with the NW/SE diagonal instead of the vertical axis.

Adjacent-numbered pages:
BP1201 BP1202 BP1203 BP1204 BP1205  *  BP1207 BP1208 BP1209 BP1210 BP1211

KEYWORD

stub, notso, stretch, left-narrow, traditional

CONCEPT symmetry_axis (info | search),
symmetry (info | search),
vertical (info | search)

AUTHOR

Aaron David Fairbanks

BP1205 Bongard Problems in which slight deformations (but perhaps across a large area) of examples can switch their sorting vs. Bongard Problems in which examples deformed slightly enough remain sorted the same way.
BP1
(edit; present; nest [left/right]; search; history)
COMMENTS

Left examples have the keyword "deformunstable" on the OEBP.

Right examples have the keyword "deformstable" on the OEBP.


For the purposes of this Bongard Problem, a "slight deformation" is a way of dragging the details of an image around which is relatively uniform in any local area and moves each point at most an arbitrarily small distance. More precise definitions could be made using mathematics.


In a "deformstable" Bongard Problem, no slight deformation should outright flip an example's sorting. It is allowed for a slight deformation to make an example sorted slightly more ambiguously.

CROSSREFS

See unstable vs. stable for changing content within a small area.

Adjacent-numbered pages:
BP1200 BP1201 BP1202 BP1203 BP1204  *  BP1206 BP1207 BP1208 BP1209 BP1210

KEYWORD

meta (see left/right), links, keyword, stability

AUTHOR

Aaron David Fairbanks

BP1204 Meta Bongard Problems of the form "arbitrarily small [transformation] applied to some examples switch their sorting vs. the sorting of each example is invariant under sufficiently small applications of [transformation]" vs. other meta Bongard Problems.
BP963
BP1205
(edit; present; nest [left/right]; search; history)
COMMENTS

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


For any "stability" Bongard Problem there could usually be made a corresponding invariance Bongard Problem ("[transformation] applied to some examples switch their sorting vs. sorting is invariant under [transformation]").


Potentially, "stability" Bongard Problems could be considered invariance Bongard Problems. On one hand, they are different, since checking whether arbitrarily small transformations switch an example's sorting is different from checking whether a particular transformation switches an example's sorting; the former is infinitely many conditions. On the other hand, there is actually only finitely much detail in any of the examples, and in practice a "stability" Bongard Problem generally just amounts to "a small application of [transformation] switches an example's sorting vs. not".

CROSSREFS

Adjacent-numbered pages:
BP1199 BP1200 BP1201 BP1202 BP1203  *  BP1205 BP1206 BP1207 BP1208 BP1209

KEYWORD

meta (see left/right), links, keyword

AUTHOR

Aaron David Fairbanks

BP1203 Bongard Problems where making a small change to some example makes it no longer fit in vs. Bongard Problems in which sufficiently small changes to examples keep them fitting in.
BP859
BP962
BP1104
BP1219
BP1
BP1220
(edit; present; nest [left/right]; search; history)
COMMENTS

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

Right-sorted Bongard Problems have the keyword "stableworld" on the OEBP.


In a "stableworld" Bongard Problem, no small change should outright make an example outright no longer fit in with the others in the Bongard Problem. It is allowed for a small change to make an example slightly less like all the others.


The meaning of "stableworld" is close to "examples have no particular format at all", but not quite the same.

CROSSREFS

See unstable vs. stable, which is about examples switching sides upon small changes instead of being rendered unsortable.

See BP1144, which is about ALL small changes to ALL examples making them unsortable.

Adjacent-numbered pages:
BP1198 BP1199 BP1200 BP1201 BP1202  *  BP1204 BP1205 BP1206 BP1207 BP1208

KEYWORD

meta (see left/right), links, keyword

AUTHOR

Aaron David Fairbanks

BP1202 Color of any region is an equal mix of the colors of the overlapping shapes vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP1197 BP1198 BP1199 BP1200 BP1201  *  BP1203 BP1204 BP1205 BP1206 BP1207

KEYWORD

precise, allsorted, notso, color

AUTHOR

Leo Crabbe

BP1201 The only triangles are the individual regions and the whole vs. there is some other triangle made of triangles.
(edit; present; nest [left/right]; search; history)
CROSSREFS

See BP1199 for the version with rectangles.

Adjacent-numbered pages:
BP1196 BP1197 BP1198 BP1199 BP1200  *  BP1202 BP1203 BP1204 BP1205 BP1206

KEYWORD

precise, traditional, left-listable, right-listable

CONCEPT triangle (info | search)

AUTHOR

Aaron David Fairbanks

BP1200 The whole rectangle can be filled in by successively replacing pairs of adjacent rectangles with one vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

Another wording: "can be repeatedly broken along 'fault lines' to yield individual pieces vs not."

REFERENCE

Robert Dawson, A forbidden suborder characterization of binarily composable diagrams in double categories, Theory and Applications of Categories, Vol. 1, No. 7, p. 146-145, 1995.

CROSSREFS

All of the examples fitting left here would fit right in BP1199 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.


All of the examples fitting right in in BP1097 (re-styled) would fit right here (besides a single solid block, but that isn't shown there).

Adjacent-numbered pages:
BP1195 BP1196 BP1197 BP1198 BP1199  *  BP1201 BP1202 BP1203 BP1204 BP1205

KEYWORD

hard, precise, challenge, proofsrequired, inductivedefinition, left-listable, right-listable

AUTHOR

Aaron David Fairbanks

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