The On-Line Encyclopedia of Bongard Problems

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Hard Bongard Problems a person has been seen to solve without cheating vs. hard Bongard Problems no one is known to have solved yet without cheating.

BP394	BP564	BP793	BP813	BP825	BP849
BP904	BP927	BP934	BP955	BP957	BP1129
BP1130

BP559	BP796	BP801	BP860	BP871	BP872
BP875	BP876	BP877	BP878	BP944	BP954
BP998	BP1011	BP1038	BP1040	BP1120	BP1200
BP1245

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	COMMENTS	`Left examples have the keyword "solved" on the OEBP.` `Right examples have the keyword "challenge" on the OEBP.`
	CROSSREFS	`Adjacent-numbered pages: BP821 BP822 BP823 BP824 BP825 * BP827 BP828 BP829 BP830 BP831`
	KEYWORD	`meta (see left/right), links, keyword, time`
	AUTHOR	`Aaron David Fairbanks`

BP825

Ticks mark an infinite sequence of angles on circle such that each angle is the double of the subsequent angle in the sequence (angle measured from rightmost indicated point) vs. not so.

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	COMMENTS	`This is solvable; it was solved by Sridhar Ramesh.` `A full turn is considered "the same angle" as no turns; likewise for adding and subtracting full turns from any angle. All sequences of angles shown start at the rightmost tick.` `It doesn't matter whether the angle is measured clockwise or counterclockwise, as long as the choice is consistent.`
	CROSSREFS	`Adjacent-numbered pages: BP820 BP821 BP822 BP823 BP824 * BP826 BP827 BP828 BP829 BP830`
	KEYWORD	`hard, convoluted, notso, math, solved`
	CONCEPT	`sequence (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP824

Objects shown chosen from collection in an ordered, algorithmic way vs. random choices involved.

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	CROSSREFS	`Adjacent-numbered pages: BP819 BP820 BP821 BP822 BP823 * BP825 BP826 BP827 BP828 BP829`
	KEYWORD	`abstract`
	AUTHOR	`Aaron David Fairbanks`

BP823

Conic section (plot of solution to conic equation) vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP818 BP819 BP820 BP821 BP822 * BP824 BP825 BP826 BP827 BP828`
	KEYWORD	`notso, math, left-couldbe`
	AUTHOR	`Aaron David Fairbanks`

BP822

Two drawn polyhedra are duals vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP817 BP818 BP819 BP820 BP821 * BP823 BP824 BP825 BP826 BP827`
	KEYWORD	`math, 3d, unorderedpair`
	AUTHOR	`Aaron David Fairbanks`

BP814

Bongard Problems with solution "object features concept: ___" vs. other BP pages.

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	COMMENTS	`Left examples have the keyword "concept" on the OEBP.` `Each concept has a corresponding metaconcept page, which describes that concept and catalogues Bongard Problems featuring that concept.`
	CROSSREFS	`Adjacent-numbered pages: BP809 BP810 BP811 BP812 BP813 * BP815 BP816 BP817 BP818 BP819`
	KEYWORD	`meta (see left/right), links, keyword`
	WORLD	`bppage [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP813

Representations of natural mathematical objects vs. representations of arbitrary objects.

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	COMMENTS	`This is a very fuzzy definition. Some left examples arguably should be placed on the right, since the particular way they are represented is arbitrary--the Platonic solids EX6730 and primes EX6734 especially, as these show arbitrary placement and arrangement of objects. Furthermore if arbitrary representations are allowed one cannot be sure for example the right hand drawing of random numbers EX6740 does not represent "numbers" in general. Still this Bongard Problem has been solved by people.`
	CROSSREFS	`Adjacent-numbered pages: BP808 BP809 BP810 BP811 BP812 * BP814 BP815 BP816 BP817 BP818`
	KEYWORD	`fuzzy, abstract, stretch, math, solved, collective, experimental, dithering`
	AUTHOR	`Aaron David Fairbanks`

BP812

Aesthetically pleasing vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP807 BP808 BP809 BP810 BP811 * BP813 BP814 BP815 BP816 BP817`
	KEYWORD	`easy, fuzzy, abstract, notso, stretch, anticomputer, subjective, invalid, experimental, funny, dithering`
	AUTHOR	`Aaron David Fairbanks`

BP811

Archimedean tiling (regular polygons, all vertices look the same) versus two-uniform tiling (regular polygons, two different kinds of vertex).

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	CROSSREFS	`Adjacent-numbered pages: BP806 BP807 BP808 BP809 BP810 * BP812 BP813 BP814 BP815 BP816`
	KEYWORD	`nice, math`
	CONCEPT	`infinite_plane (info \| search),` `tiling (info \| search),` `symmetry (info \| search)`
	WORLD	`wallpaper_tiling [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP810

Figures can be transformed into one another by smooth stretching (intersection points stay constant; paths connecting those points remain), while remaining within the 2d box vs. movement out of the plane required.

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	CROSSREFS	`All examples here fit left in BP809, a version where the figures are allowed to pass through themselves while being deformed.` `Adjacent-numbered pages: BP805 BP806 BP807 BP808 BP809 * BP811 BP812 BP813 BP814 BP815`
	KEYWORD	`nice, math, unorderedpair, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`two_homeomorphic_figures_made_of_curves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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