The On-Line Encyclopedia of Bongard Problems

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Odd number of dots vs. even number of dots.

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	CROSSREFS	`See BP334 for a version of the same idea, but using arbitrary shapes instead of dots.` `Adjacent-numbered pages: BP329 BP330 BP331 BP332 BP333 * BP335 BP336 BP337 BP338 BP339`
	KEYWORD	`precise, allsorted, number, math, left-narrow, right-narrow, right-null, help, traditional, preciseworld`
	CONCEPT	`even_odd (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

One object does not belong to the pattern of the rest vs. all objects form one pattern.

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	COMMENTS	`The left examples are the right examples with one object altered, which makes the solution easier to see.` `"Odd one out."`
	CROSSREFS	`Adjacent-numbered pages: BP344 BP345 BP346 BP347 BP348 * BP350 BP351 BP352 BP353 BP354`
	KEYWORD	`anticomputer, help, contributepairs, traditional, rules`
	CONCEPT	`categorization (info \| search),` `existence (info \| search),` `feature_cluster (info \| search),` `number_cluster (info \| search),` `shape_cluster (info \| search),` `cluster_of_one (info \| search),` `cluster (info \| search),` `one (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

No knot (unknot) vs. knot.

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	COMMENTS	`To hint at the solution, left examples can be arranged in a sequence to show the unknot tying itself.`
	CROSSREFS	`Adjacent-numbered pages: BP377 BP378 BP379 BP380 BP381 * BP383 BP384 BP385 BP386 BP387`
	KEYWORD	`nice, math, anticomputer, help, traditional`
	CONCEPT	`knot (info \| search),` `topological_transformation (info \| search)`
	WORLD	`knot [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Square number of dots vs. non-square number of dots.

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	COMMENTS	`All examples in this Problem are a collection of dots.` `An equivalent solution is "Dots can be arranged into a square lattice whose convex hull is a square vs. not so". - Leo Crabbe, Aug 01 2020`
	CROSSREFS	`Adjacent-numbered pages: BP379 BP380 BP381 BP382 BP383 * BP385 BP386 BP387 BP388 BP389`
	EXAMPLE	`A single dot fits because 1 = 11.` `A pair of dots does not fit because there is no integer x such that 2 = xx.`
	KEYWORD	`nice, precise, allsorted, number, math, left-narrow, left-null, help, traditional, preciseworld`
	CONCEPT	`square_number (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

Triangular number of dots vs. non-triangular number of dots

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	COMMENTS	`All examples in this Problem are groups of black dots.` `The nth triangular number is the sum over the natural numbers from 1 to n, where n > 0. Note: 0 is the 0th triangular number. The first few triangular numbers are 0, 1, 3 (= 1+2) and 6 (= 1+2+3)`
	CROSSREFS	`Adjacent-numbered pages: BP564 BP565 BP566 BP567 BP568 * BP570 BP571 BP572 BP573 BP574`
	KEYWORD	`nice, precise, allsorted, notso, number, math, left-narrow, left-null, help, preciseworld`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

Image of a Bongard Problem with no simple solution versus image of a Bongard Problem with a simple solution.

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	COMMENTS	`Left examples have no solution, but they do not break the rules in ways so extreme that it is plainly impossible for them to have a solution, such as including the same image on both sides or including no images per side. (See such as including the same image on both sides or including no images per side.`
	CROSSREFS	`See BP522 for the version with links to pages on the OEBP instead of images of Bongard Problems (miniproblems).` `See BP968 (flipped) for a version of this Bongard Problem including examples of invalid Bongard Problems that don't even admit a convoluted solution (the same image appears on both sides).` `Also see BP1080, which is similar to BP968, but including various different formats of Bongard Problems, distinguishing them from arbitrary images that are not Bongard Problems.` `Adjacent-numbered pages: BP824 BP825 BP826 BP827 BP828 * BP830 BP831 BP832 BP833 BP834`
	KEYWORD	`nice, meta (see left/right), miniproblems, creativeexamples, left-unknowable, right-narrow, assumesfamiliarity, structure, help, presentationinvariant`
	CONCEPT	`existence (info \| search),` `simplicity (info \| search),` `zero (info \| search)`
	WORLD	`boxes_bpimage_three_per_side_nosoln_allowed [smaller \| same \| bigger] zoom in right (boxes_bpimage_three_per_side)`
	AUTHOR	`Aaron David Fairbanks`

Black shapes can be arranged such that they fit inside rectangular outline vs. not so.

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	COMMENTS	`There is a slight ambiguity here regarding whether a shape could be placed within another shape's hole. This is a question of how one perceives the Problem: are we sliding shapes around on a table in 2D or are we allowed to 'lift' them in 3D space?`
	CROSSREFS	`Adjacent-numbered pages: BP887 BP888 BP889 BP890 BP891 * BP893 BP894 BP895 BP896 BP897`
	KEYWORD	`nice, precise, perfect, pixelperfect, help`
	CONCEPT	`rotation_required (info \| search),` `physically_fitting (info \| search)`
	AUTHOR	`Leo Crabbe`

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

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	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`

Number of dots is a power of 2 vs. not so.

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	COMMENTS	`Numbers of dots on the left can be obtained by repeatedly doubling 1 dot.` `Numbers of dots on the left are the number of corners of a cube in some dimension.`
	CROSSREFS	`Adjacent-numbered pages: BP983 BP984 BP985 BP986 BP987 * BP989 BP990 BP991 BP992 BP993`
	KEYWORD	`stub, precise, allsorted, number, left-narrow, right-null, help, preciseworld`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

Number of dots is n factorial for some n vs. not so.

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	COMMENTS	`Zero is intentionally left out to avoid confusion (although it would fit right).`
	CROSSREFS	`Adjacent-numbered pages: BP984 BP985 BP986 BP987 BP988 * BP990 BP991 BP992 BP993 BP994`
	KEYWORD	`stub, precise, number, math, left-narrow, right-null, help, preciseworld`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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