The On-Line Encyclopedia of Bongard Problems

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Image of Bongard Problem that would sort ANY image of a valid Bongard Problem on one of its sides vs. image of Bongard Problem whose categorization of a BP image would depend on the solution or examples in it.

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	COMMENTS	`"Any" here means any image of a Bongard Problem in the relevant format, i.e. with white background, black vertical dividing line, and examples in boxes on either side.` `All examples shown in this Problem clearly sort themselves on the left or right.` `A self-referential but maybe simpler solution is "would sort all examples in this whole Bongard Problem on one of its sides vs. not so." Users adding examples please try to maintain this: for any example you add to the right of this Bongard Problem, make sure it does not sort all the other examples in this Bongard Problem on just one of its sides. - Aaron David Fairbanks, Aug 26 2020`
	CROSSREFS	`Adjacent-numbered pages: BP939 BP940 BP941 BP942 BP943 * BP945 BP946 BP947 BP948 BP949`
	KEYWORD	`hard, challenge, presentationinvariant`
	WORLD	`boxes_bpimage_sorts_self [smaller \| same \| bigger] zoom in left \| zoom in right`
	AUTHOR	`Jago Collins`

BP945

Cube number of dots vs. non-cube number of dots.

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

	CROSSREFS	`Adjacent-numbered pages: BP940 BP941 BP942 BP943 BP944 * BP946 BP947 BP948 BP949 BP950`
	KEYWORD	`precise, allsorted, number, left-null, help, preciseworld`
	CONCEPT	`cube (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP946

Can be constructed using 2 identical copies of an image (full overlapping not allowed) vs. not so.

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	COMMENTS	`"Full overlapping not allowed" means you cannot overlay an image onto itself without moving it; if this were allowed all images would be sorted on the left. The copies can be moved around (translated) in 2D but can not be flipped or rotated.` `There are examples on the right drawn with thick lines, and these could be created by copying an image with slightly thinner lines and moving it over a tiny amount. If you fix this issue by saying "the copy has to be moved over more than a tiny amount" then the Bongard Problem is perfect but not precise, but if you fix this issue by saying "interpret the figures as made up of (infinitesimally) thin lines" then it's precise but not perfect. - Aaron David Fairbanks, Jun 17 2023`
	CROSSREFS	`Adjacent-numbered pages: BP941 BP942 BP943 BP944 BP945 * BP947 BP948 BP949 BP950 BP951`
	KEYWORD	`nice, notso, creativeexamples`
	AUTHOR	`Leo Crabbe`

BP949

Two unique distances between points vs. three unique distances between points.

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	CROSSREFS	`Adjacent-numbered pages: BP944 BP945 BP946 BP947 BP948 * BP950 BP951 BP952 BP953 BP954`
	KEYWORD	`nice, precise, allsorted, stretch, perfect, traditional, preciseworld`
	CONCEPT	`two (info \| search),` `three (info \| search)`
	WORLD	`3_or_4_points [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP951

Process described leaves some inputs invariant vs. no output will resemble its input.

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	COMMENTS	`There are many ambiguities here. The solver is expected to determine what things are "allowed" to be inputs for each process. To avoid confusion examples should not be sorted differently if you consider inputting nothing.` `In each example there is at least some overlap between the set of possible inputs and the set of possible outputs for each process. If we did not apply this constraint, an easy example to be sorted right would be a process that turns blue shapes red.` `A harder-to-read but more clearly defined version of this Problem could include within each example a mini Bongard Problem sorting left all allowed inputs for the process.`
	REFERENCE	`https://en.wikipedia.org/wiki/Fixed_point_(mathematics)`
	CROSSREFS	`Adjacent-numbered pages: BP946 BP947 BP948 BP949 BP950 * BP952 BP953 BP954 BP955 BP956`
	KEYWORD	`structure, rules, miniworlds`
	CONCEPT	`function (info \| search)`
	AUTHOR	`Leo Crabbe`

BP956

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

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	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`

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`

BP962

White vs. black.

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	CROSSREFS	`Adjacent-numbered pages: BP957 BP958 BP959 BP960 BP961 * BP963 BP964 BP965 BP966 BP967`
	KEYWORD	`precise, allsorted, minimal, dual, blackwhite, gap, left-finite, right-finite, left-full, right-full, left-null, finished, preciseworld, unstableworld`
	WORLD	`[smaller \| same \| bigger] zoom in left (blank_image) \| zoom in right (black_image)`
	AUTHOR	`Leo Crabbe`

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.

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	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`

BP966

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

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	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`

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