The On-Line Encyclopedia of Bongard Problems

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Triangle is smallest black shape vs. square is smallest black shape.

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

	CROSSREFS	`Adjacent-numbered pages: BP964 BP965 BP966 BP967 BP968 * BP970 BP971 BP972 BP973 BP974`
	KEYWORD	`stub`
	CONCEPT	`size (info \| search),` `square (info \| search),` `triangle (info \| search),` `most (info \| search),` `quantity_comparison (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP966

Even number of white regions vs. odd number of white regions.

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

	COMMENTS	`See BP889 for the version in which the background doesn't count.`
	CROSSREFS	`Adjacent-numbered pages: BP961 BP962 BP963 BP964 BP965 * BP967 BP968 BP969 BP970 BP971`
	KEYWORD	`nice, boundingbox, number, right-null, perfect, pixelperfect, traditional, bordercontent`
	CONCEPT	`separated_regions (info \| search),` `even_odd (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP965

If you place the image on top of itself so that it lines up with itself exactly within a small region, it also lines up everywhere else vs. not so.

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

	COMMENTS	`Rotations are allowed. To avoid confusion about whether reflections are allowed, no examples are included on the right that require reflections to match up with themselves locally but not globally; no examples are included on the left that can match up with themselves locally but not globally using a reflection.` `Only parts of ellipses are used, and only one type of ellipse per image, to make everything easier to read and reason about.`
	CROSSREFS	`See BP1246 for a variation on this idea where instead of lining the image up with itself along arbitrarily small regions, you line the image up with itself along individual separate objects.` `Adjacent-numbered pages: BP960 BP961 BP962 BP963 BP964 * BP966 BP967 BP968 BP969 BP970`
	KEYWORD	`hard, precise, distractingworld, perfect`
	CONCEPT	`local_global (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP961

Includes itself on the left vs. includes itself on the right.

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

	COMMENTS	`Some examples are Bongard Problems with this solution.`
	CROSSREFS	`Adjacent-numbered pages: BP956 BP957 BP958 BP959 BP960 * BP962 BP963 BP964 BP965 BP966`
	KEYWORD	`nice, precise, dual, handed, leftright, perfect, infinitedetail, both, neither, preciseworld`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP959

This image of this Bongard Problem vs. empty image.

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

	CROSSREFS	`See BP953, BP902.` `Adjacent-numbered pages: BP954 BP955 BP956 BP957 BP958 * BP960 BP961 BP962 BP963 BP964`
	KEYWORD	`less, meta (see left/right), miniproblems, left-finite, right-finite, left-full, right-full, right-null, perfect, infinitedetail, finished, experimental, funny`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search)`
	WORLD	`zoom in left \| zoom in right (blank_image)`
	AUTHOR	`Aaron David Fairbanks, Leo Crabbe`

BP956

Nested pairs of brackets vs. other arrangement of brackets (some open brackets are not closed or there are extra closing brackets).

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

	COMMENTS	`Examples on the left are also known as "Dyck words".`
	REFERENCE	`https://en.wikipedia.org/wiki/Dyck_language`
	CROSSREFS	`Adjacent-numbered pages: BP951 BP952 BP953 BP954 BP955 * BP957 BP958 BP959 BP960 BP961`
	KEYWORD	`easy, nice, precise, allsorted, unwordable, notso, sequence, traditional, inductivedefinition, preciseworld, left-listable, right-listable`
	CONCEPT	`recursion (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP932

Every vertex is connected to every other vs. vertices are connected in a cycle (no other connections).

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

	COMMENTS	`Complete graphs with zero, one, two, or three vertices would be ambiguously categorized (fit in overlap of both sides).` `Left examples are called "fully connected graphs." Right examples are called "cycle graphs."`
	CROSSREFS	`Adjacent-numbered pages: BP927 BP928 BP929 BP930 BP931 * BP933 BP934 BP935 BP936 BP937`
	KEYWORD	`precise, left-narrow, right-narrow, both, preciseworld`
	CONCEPT	`graph (info \| search),` `distinguishing_crossing_curves (info \| search),` `all (info \| search),` `loop (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP931

Some number labels its own position in the sequence from left to right vs. not so.

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

	COMMENTS	`Right examples are called "derangements".`
	CROSSREFS	`Adjacent-numbered pages: BP926 BP927 BP928 BP929 BP930 * BP932 BP933 BP934 BP935 BP936`
	KEYWORD	`handed, leftright, sequence, traditional, left-listable, right-listable`
	CONCEPT	`number (info \| search),` `dot (info \| search),` `self-reference (info \| search)`
	WORLD	`dot_clusters_sequence_horizontal [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP928

All subsets of a collection vs. not.

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

	CROSSREFS	`Adjacent-numbered pages: BP923 BP924 BP925 BP926 BP927 * BP929 BP930 BP931 BP932 BP933`
	KEYWORD	`notso, math, traditional`
	AUTHOR	`Aaron David Fairbanks`

BP927

Image of Bongard Problem whose self-sorting depends on examples in it vs. image of Bongard Problem that will sort any Bongard Problem with its solution on either its left or right regardless of examples chosen.

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

	COMMENTS	`All examples are Bongard Problems fitting left or right in BP793.` `All examples here are in the conventional format, i.e. white background, black vertical dividing line, and examples in boxes on either side.` `Border cases are Bongard Problems that always self-sort one way given their particular visual format (e.g. fixed number of boxes), but self-sort a different way in another slightly different format.` `Meta Bongard Problems appearing in BP793 that are presentationinvariant necessarily fit right here.` `It is interesting to think about how this Bongard Problem sorts itself. The only self-consistent answer is that it fits right.`
	CROSSREFS	`See BP793 "sorts self left vs. sorts self right".` `See BP944 "sorts every BP on one side vs. doesn't".` `Adjacent-numbered pages: BP922 BP923 BP924 BP925 BP926 * BP928 BP929 BP930 BP931 BP932`
	KEYWORD	`hard, solved, presentationinvariant, visualimagination`
	WORLD	`boxes_bpimage_sorts_self [smaller \| same \| bigger] zoom in left (boxes_bpimage_sorts_self_incarnation_dependent) \| zoom in right`
	AUTHOR	`Aaron David Fairbanks`

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