The On-Line Encyclopedia of Bongard Problems

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Empty image vs. non-empty image.

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	COMMENTS	`The first Bongard Problem.` `All examples in this Bongard Problem are line drawings (one or more connected figures made up of curved and non-curved lines).`
	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 214.`
	CROSSREFS	`Adjacent-numbered pages: * BP2 BP3 BP4 BP5 BP6`
	EXAMPLE	`A circle fits on the right because it is not nothing.`
	KEYWORD	`easy, nice, precise, allsorted, unstable, world, left-narrow, left-finite, left-full, left-null, perfect, pixelperfect, finished, traditional, stableworld, deformstable, bongard`
	CONCEPT	`empty (info \| search),` `existence (info \| search),` `zero (info \| search)`
	WORLD	`zoom in left (blank_image) \| zoom in right (curves_drawing)`
	AUTHOR	`Mikhail M. Bongard`

Dots equal in number to the sides of the closed region vs. dots unequal in number to the sides of the closed region.

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	CROSSREFS	`Adjacent-numbered pages: BP132 BP133 BP134 BP135 BP136 * BP138 BP139 BP140 BP141 BP142`
	KEYWORD	`left-null, traditional`
	CONCEPT	`same_number (info \| search),` `same (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

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, collection`
	CONCEPT	`square_number (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

Some zoomed-in (cropped) version of an image of a hollow circle vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP520 BP521 BP522 BP523 BP524 * BP526 BP527 BP528 BP529 BP530`
	KEYWORD	`stretch, boundingbox, left-null, traditional, left-couldbe`
	CONCEPT	`circle (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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`

Rows show all possible ways a certain number of dots can be divided between a certain number of bins vs. not so.

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	COMMENTS	`The rows in the panels on the right hand side show all the ways you can divide a certain number of dots between a certain number of bins, ignoring which bins they are placed in.`
	CROSSREFS	`Adjacent-numbered pages: BP899 BP900 BP901 BP902 BP903 * BP905 BP906 BP907 BP908 BP909`
	KEYWORD	`solved, left-null, grid, left-listable, right-listable`
	CONCEPT	`permutation (info \| search)`
	AUTHOR	`Molly C Klenzak`

Graph can be redrawn such that no edges intersect vs. not so.

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	COMMENTS	`A graph is a collection of vertices and edges. Vertices are the dots and edges are the lines that connect the dots. On the left, all edges can be redrawn (curved lines are allowed and moving vertices is allowed) such that no edges cross each other and each vertex is still connected to the same other vertices. These graphs are called planar.`
	CROSSREFS	`Adjacent-numbered pages: BP900 BP901 BP902 BP903 BP904 * BP906 BP907 BP908 BP909 BP910`
	KEYWORD	`nice, precise, allsorted, notso, math, left-null, preciseworld`
	CONCEPT	`graph (info \| search),` `topological_transformation (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak`

Finite number of dots vs. infinite number of dots.

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	CROSSREFS	`Adjacent-numbered pages: BP910 BP911 BP912 BP913 BP914 * BP916 BP917 BP918 BP919 BP920`
	KEYWORD	`less, notso, spectrum, number, example, left-null, impossible, experimental`
	CONCEPT	`finite_infinite (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

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`

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`

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