The On-Line Encyclopedia of Bongard Problems

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BP1

Empty image vs. non-empty image.

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	COMMENTS	`The first Bongard Problem.` `All examples in this Bongard Problem are line drawings (one or more connected figures made up of curved and non-curved lines).`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 214.`
	CROSSREFS	`Adjacent-numbered pages: * BP2 BP3 BP4 BP5 BP6`
	EXAMPLE	`A circle fits on the right because it is not nothing.`
	KEYWORD	`easy, nice, precise, allsorted, unstable, world, left-narrow, left-finite, left-full, left-null, perfect, pixelperfect, finished, traditional, stableworld, deformstable, bongard`
	CONCEPT	`empty (info \| search),` `existence (info \| search),` `zero (info \| search)`
	WORLD	`zoom in left (blank_image) \| zoom in right (curves_drawing)`
	AUTHOR	`Mikhail M. Bongard`

BP2

Big vs. small.

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	COMMENTS	`The meaning of "big" left intentionally vague. There are various specific ways to define size, such as diameter, minimum distance between points on edge, and size of smallest bounding circle.` `All examples in this Bongard Problem are single simple shapes, either outlines or solid black.` `All examples on the same side are approximately the same size.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 214.`
	CROSSREFS	`Adjacent-numbered pages: BP1 * BP3 BP4 BP5 BP6 BP7`
	KEYWORD	`easy, nice, fuzzy, spectrum, size, stable, finished, traditional, continuous, bongard`
	CONCEPT	`size (info \| search)`
	WORLD	`outline_or_fill_shape [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

BP3

Hollow outline vs. filled in solid.

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	COMMENTS	`All examples in this Bongard Problem are single simple shapes.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 214.`
	CROSSREFS	`Adjacent-numbered pages: BP1 BP2 * BP4 BP5 BP6 BP7 BP8`
	KEYWORD	`easy, nice, precise, allsorted, world, gap, finished, traditional, preciseworld, bongard`
	CONCEPT	`outlined_filled (info \| search),` `texture (info \| search)`
	WORLD	`outline_or_fill_shape [smaller \| same \| bigger] zoom in left (shape_outline) \| zoom in right (fill_shape)`
	AUTHOR	`Mikhail M. Bongard`

BP4

Convex vs. concave.

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	COMMENTS	`All examples in this Problem are outlines of shapes or solid black shapes.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 215.`
	CROSSREFS	`BP136 is the same solution (flipped) but with only polygonal outlines and also with extraneous dots distracting from the solution.` `Adjacent-numbered pages: BP1 BP2 BP3 * BP5 BP6 BP7 BP8 BP9`
	KEYWORD	`easy, nice, precise, unstable, right-narrow, finished, traditional, bongard`
	CONCEPT	`concave_convex_angle (info \| search)`
	WORLD	`outline_or_fill_shape [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

BP5

Is polygon vs. is smooth without straight lines or corners.

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	COMMENTS	`All examples in this Bongard Problem are outlines of shapes.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 215.`
	CROSSREFS	`Adjacent-numbered pages: BP1 BP2 BP3 BP4 * BP6 BP7 BP8 BP9 BP10`
	KEYWORD	`easy, nice, ignoreimperfections, finished, traditional, bongard`
	CONCEPT	`curved_straight (info \| search)`
	WORLD	`shape_outline [smaller \| same \| bigger] zoom in left (polygon_outline)`
	AUTHOR	`Mikhail M. Bongard`

BP6

Triangle vs. quadrilateral.

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	COMMENTS	`All examples in this Problem are outlines of polygons or solid black polygons.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 215.`
	CROSSREFS	`BP1211 is "triangle vs. anything else".` `Adjacent-numbered pages: BP1 BP2 BP3 BP4 BP5 * BP7 BP8 BP9 BP10 BP11`
	KEYWORD	`easy, nice, precise, number, ignoreimperfections, finished, traditional, preciseworld, bongard`
	CONCEPT	`number (info \| search),` `triangle (info \| search),` `three (info \| search),` `four (info \| search)`
	WORLD	`Multiple options: polygon_outline_or_fill [smaller \| same \| bigger], triangle_or_quadrilateral_outline_or_fill [smaller \| same \| bigger]`
	AUTHOR	`Mikhail M. Bongard`

BP986

Palindromes vs. not palindromes.

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	COMMENTS	`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."`
	CROSSREFS	`Adjacent-numbered pages: BP981 BP982 BP983 BP984 BP985 * BP987 BP988 BP989 BP990 BP991`
	KEYWORD	`nice, precise, allsorted, notso, sequence, traditional`
	CONCEPT	`element_wise_symmetry (info \| search),` `identical (info \| search),` `sequence (info \| search),` `same_shape (info \| search),` `same (info \| search),` `symmetry (info \| search)`
	AUTHOR	`Jago Collins`

BP1197

No sequence is repeated twice in a row vs. some sequence is repeated twice in a row.

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	REFERENCE	`https://en.wikipedia.org/wiki/Square-free_word`
	CROSSREFS	`Adjacent-numbered pages: BP1192 BP1193 BP1194 BP1195 BP1196 * BP1198 BP1199 BP1200 BP1201 BP1202`
	KEYWORD	`precise, allsorted, notso, left-narrow, traditional, preciseworld, dithering, left-listable`
	CONCEPT	`two (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1199

The only rectangles are the individual regions and the whole vs. there is some other rectangle made of rectangles.

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CROSSREFS

All of the examples fitting left here would fit right in BP1200 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.

All of the examples fitting left in BP1097 (re-styled) would fit right here (besides the two possible arrangements made up of just two rectangles, but those aren't shown there).

See BP1201 for the version with triangles.

Adjacent-numbered pages: BP1194 BP1195 BP1196 BP1197 BP1198 * BP1200 BP1201 BP1202 BP1203 BP1204

KEYWORD

precise, traditional, left-listable, right-listable

AUTHOR

Aaron David Fairbanks

BP1200

The whole rectangle can be filled in by successively replacing pairs of adjacent rectangles with one vs. not so.

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	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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