The On-Line Encyclopedia of Bongard Problems

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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.

BP1

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	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`

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

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	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`

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

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	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`

BP1206

Vertical axis of symmetry vs. no vertical axis of symmetry.

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	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`

BP1207

Horizontal axis of symmetry vs. no horizontal axis of symmetry.

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	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`

BP1208

More triangles left than right vs. more triangles right than left.

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	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`

BP1209

Empty square present vs. no square present.

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	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`

BP1210

Vertical, horizontal, and both diagonal axes of symmetry vs. not (more specifically, one or fewer axes of symmetry).

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	CROSSREFS	`Adjacent-numbered pages: BP1205 BP1206 BP1207 BP1208 BP1209 * BP1211 BP1212 BP1213 BP1214 BP1215`
	KEYWORD	`stub, left-narrow, traditional, blackwhiteinvariant`
	CONCEPT	`symmetry_axis (info \| search),` `symmetry (info \| search),` `four (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1211

Triangle vs. anything else.

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	CROSSREFS	`See BP6 for "triangle vs. quadrilateral".` `Adjacent-numbered pages: BP1206 BP1207 BP1208 BP1209 BP1210 * BP1212 BP1213 BP1214 BP1215 BP1216`
	KEYWORD	`stub, notso, left-narrow, traditional`
	CONCEPT	`triangle (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1212

Shape tiles a square vs. not

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	CROSSREFS	`Adjacent-numbered pages: BP1207 BP1208 BP1209 BP1210 BP1211 * BP1213 BP1214 BP1215 BP1216 BP1217`
	KEYWORD	`precise`
	WORLD	`fill_shape [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

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