The On-Line Encyclopedia of Bongard Problems

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Figure with points (small white circles) can be smoothly deformed within the 2D plane without passing through itself so that all points touch to make the other figure vs. not so (movement out of the plane required).

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	CROSSREFS	`All examples here fit left in BP369, a version where the figure is allowed to pass through itself while being deformed.` `Adjacent-numbered pages: BP846 BP847 BP848 BP849 BP850 * BP852 BP853 BP854 BP855 BP856`
	KEYWORD	`math, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`figure_made_of_curves_and_quotient_by_hollow_dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1080

Image of a Bongard Problem vs. other image.

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	COMMENTS	`Various formats of Bongard Problems frequently seen on the OEBP are showcased on the left side here.`
	CROSSREFS	`See BP968, which distinguishes solvable Bongard Problems from other images still formatted like Bongard Problems (as opposed to this page, which distinguishes Bongard Problems from any other images.)` `Adjacent-numbered pages: BP1075 BP1076 BP1077 BP1078 BP1079 * BP1081 BP1082 BP1083 BP1084 BP1085`
	KEYWORD	`notso, teach, left-narrow, right-null`
	CONCEPT	`bongard_problem (info \| search),` `categorization (info \| search)`
	AUTHOR	`Leo Crabbe`

BP20

Both dots touching same bulb vs. dots on opposite bulbs.

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	COMMENTS	`All examples in this Bongard Problem are outlines of smooth shapes with a pinch in the middle and with two tiny black circles attached to the outside of the shape (tangent).` `A shape with either dot tangent to the the valley section would be ambiguous.`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 220.`
	CROSSREFS	`Adjacent-numbered pages: BP15 BP16 BP17 BP18 BP19 * BP21 BP22 BP23 BP24 BP25`
	KEYWORD	`nice, finished, traditional, bongard`
	CONCEPT	`near_far (info \| search),` `neck_narrowing (info \| search),` `separation_of_joined_objects (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

BP1248

No straight path passes from a white region to a grey one, then to white again (without hitting black) vs. such a path exists.

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	CROSSREFS	`Any small amount of convexity along the grey border will cause an example to fit right.` `Adjacent-numbered pages: BP1243 BP1244 BP1245 BP1246 BP1247 * BP1249 BP1250 BP1251 BP1252 BP1253`
	KEYWORD	`precise, allsorted, unwordable, notso, perfect`
	CONCEPT	`imagined_line_or_curve (info \| search)`
	WORLD	`[smaller \| same \| bigger] zoom in left`
	AUTHOR	`Leo Crabbe`

BP141

Two clusters of three and two objects each vs. one cluster of three objects and two different objects.

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	CROSSREFS	`Adjacent-numbered pages: BP136 BP137 BP138 BP139 BP140 * BP142 BP143 BP144 BP145 BP146`
	KEYWORD	`traditional`
	CONCEPT	`and (info \| search),` `number (info \| search),` `feature_cluster (info \| search),` `cluster (info \| search),` `same_shape (info \| search),` `same (info \| search),` `one (info \| search),` `two (info \| search),` `three (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

BP997

There exists a loop that passes through every white square once without passing through the black square vs. there exists no such loop.

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	CROSSREFS	`Adjacent-numbered pages: BP992 BP993 BP994 BP995 BP996 * BP998 BP999 BP1000 BP1001 BP1002`
	KEYWORD	`precise, allsorted, grid, preciseworld, left-listable, right-listable`
	CONCEPT	`path (info \| search)`
	AUTHOR	`James Tanton`

BP230

The sum of vertices, line ends, and dots is 7 vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP225 BP226 BP227 BP228 BP229 * BP231 BP232 BP233 BP234 BP235`
	KEYWORD	`arbitrary, traditional`
	CONCEPT	`addition (info \| search),` `line_or_curve_endpoint (info \| search),` `dot (info \| search),` `vertex_of_meeting_lines (info \| search)`
	AUTHOR	`Giuseppe Insana`

BP987

Solution could appear in a Bongard Problem featuring an image of itself on either of its sides vs. solution can appear in a Bongard Problem featuring an image of itself on a certain side only.

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	COMMENTS	`All examples are Bongard Problems fitting left in BP954.` `This is very close to BP927, specialized to Bongard Problems fitting left in BP954. The difference is that a Bongard Problem solution would fit left in BP927 but right here if it can sort images of it on both sides, but it is impossible to make an image of it fractally including itself on a certain side. An example is EX7997.` `Meta Bongard Problems appearing in BP793 that are presentationinvariant necessarily fit right here.`
	CROSSREFS	`Adjacent-numbered pages: BP982 BP983 BP984 BP985 BP986 * BP988 BP989 BP990 BP991 BP992`
	KEYWORD	`abstract, meta (see left/right), miniproblems, creativeexamples, assumesfamiliarity, structure, presentationinvariant, visualimagination`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	AUTHOR	`Leo Crabbe`

BP897

Wide angles connected to narrow angles vs. not so.

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	COMMENTS	`Another solution is that right examples can be folded down flat onto one isosceles triangle while left examples cannot.` `All examples in this Problem feature four isosceles triangles connected by corners and/or edges.`
	CROSSREFS	`This was conceived as a false solution for BP898.` `Adjacent-numbered pages: BP892 BP893 BP894 BP895 BP896 * BP898 BP899 BP900 BP901 BP902`
	KEYWORD	`precise, allsorted, notso, traditional, preciseworld`
	CONCEPT	`triangle (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak, Aaron David Fairbanks`

BP114

Four X-like points vs. two X-like points.

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	CROSSREFS	`Adjacent-numbered pages: BP109 BP110 BP111 BP112 BP113 * BP115 BP116 BP117 BP118 BP119`
	KEYWORD	`traditional`
	CONCEPT	`x-crossing (info \| search),` `number (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

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