The On-Line Encyclopedia of Bongard Problems

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As one quantity increases an equally obvious opposite quantity decreases vs. there is only one obvious quantity, which increases as the sequence progresses right.

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	COMMENTS	`Another way of phrasing the solution: "Neither direction would more naturally be called increase in quantity vs. rightward progression would be called an increase."` `Most right examples shown are unboundedly increasing, since finite sequences showing a quantity increasing usually also suggest "distance to end of sequence" as a decreasing opposite quantity. Even so, there are some finite sequences with one direction more intuitively increase-like than the other.`
	CROSSREFS	`Adjacent-numbered pages: BP888 BP889 BP890 BP891 BP892 * BP894 BP895 BP896 BP897 BP898`
	KEYWORD	`creativeexamples, structure, rules`
	WORLD	`constant_change_seq_increase_right [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP870

Increasing quantity has upper bound (will get "stopped" by something) vs. not so.

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	COMMENTS	`Related to BP380.`
	CROSSREFS	`Adjacent-numbered pages: BP865 BP866 BP867 BP868 BP869 * BP871 BP872 BP873 BP874 BP875`
	KEYWORD	`creativeexamples, structure, rules`
	WORLD	`constant_change_seq_increase_right [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP859

Black pixel vs. white pixel.

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	COMMENTS	`This is the "smallest" possible valid Bongard Problem on black and white bitmap images (up to switching sides).`
	CROSSREFS	`Adjacent-numbered pages: BP854 BP855 BP856 BP857 BP858 * BP860 BP861 BP862 BP863 BP864`
	KEYWORD	`minimal, blackwhite, perfect, pixelperfect, experimental, funny, unstableworld`
	WORLD	`black_or_white_pixel [smaller \| same \| bigger] zoom in left (black_pixel) \| zoom in right (white_pixel)`
	AUTHOR	`Aaron David Fairbanks`

BP855

Object below ambiguously sorted (not clearly left or right) by Bongard Problem image above vs. object below clearly sorted by Bongard Problem image above.

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	COMMENTS	`Represented are Bongard Problems and an extra object to sort, a.k.a. the Bongard's Dozen format.` `A variety of kinds of ambiguities showcased as left examples.`
	CROSSREFS	`Adjacent-numbered pages: BP850 BP851 BP852 BP853 BP854 * BP856 BP857 BP858 BP859 BP860`
	KEYWORD	`nice, creativeexamples, structure, rules`
	WORLD	`bdimage_shapes_extra_below [smaller \| same \| bigger] zoom in right (bdimage_shapes_extra_below_clear_sort)`
	AUTHOR	`Aaron David Fairbanks`

BP854

Nothing vs. nothing.

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	CROSSREFS	`Adjacent-numbered pages: BP849 BP850 BP851 BP852 BP853 * BP855 BP856 BP857 BP858 BP859`
	KEYWORD	`left-finite, right-finite, left-full, right-full, finished, experimental, funny`
	WORLD	`nothing [same \| bigger] zoom in left (nothing) \| zoom in right (nothing)`
	AUTHOR	`Aaron David Fairbanks`

BP853

Prime knot vs. composite knot.

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	REFERENCE	`https://en.wikipedia.org/wiki/Prime_knot`
	CROSSREFS	`Adjacent-numbered pages: BP848 BP849 BP850 BP851 BP852 * BP854 BP855 BP856 BP857 BP858`
	KEYWORD	`math, hardsort`
	CONCEPT	`knot (info \| search)`
	WORLD	`knot [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP852

Object shown below is the "limit" of the sequence above (end result after "infinite time") versus not so.

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	COMMENTS	`The conceptual limit of the sequence may not be the limit of the points in the image. For example in a sequence of halvings the limit value is never reached, so the bottom would never change color and thus its limit would not would not either.` `Sequences progress from left to right (and there is not usually a way to intuitively extend the sequence in the other direction).`
	CROSSREFS	`Adjacent-numbered pages: BP847 BP848 BP849 BP850 BP851 * BP853 BP854 BP855 BP856 BP857`
	KEYWORD	`notso, creativeexamples, perfect, infinitedetail, assumesfamiliarity, structure, contributepairs, rules`
	AUTHOR	`Aaron David Fairbanks`

BP851

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`

BP849

Loop is time-symmetrical (loop is like ABCBABCBA... instead of like ABCABCABC...) up to rotation of object vs. not so (neither up to rotation nor reflection).

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`Adjacent-numbered pages: BP844 BP845 BP846 BP847 BP848 * BP850 BP851 BP852 BP853 BP854`
	KEYWORD	`hard, convoluted, solved, rules`
	WORLD	`constant_change_seq_loop_right [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP848

Lower bound object shown vs. not so.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`Adjacent-numbered pages: BP843 BP844 BP845 BP846 BP847 * BP849 BP850 BP851 BP852 BP853`
	KEYWORD	`creativeexamples, structure, contributepairs, rules`
	WORLD	`constant_change_seq_increase_right_low_bound_obj [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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