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`

One outer outline vs. more than one outer outline.

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	CROSSREFS	`Adjacent-numbered pages: BP317 BP318 BP319 BP320 BP321 * BP323 BP324 BP325 BP326 BP327`
	KEYWORD	`nice, precise, allsorted, traditional`
	CONCEPT	`separated_regions (info \| search),` `interior_exterior (info \| search),` `outer_outline (info \| search),` `recursion (info \| search),` `one (info \| search)`
	WORLD	`nest_shapes [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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`

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`

Can be cut into tiles forming a checkerboard pattern vs. not so.

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	COMMENTS	`All examples in this Problem are grids consisting of two objects.`
	CROSSREFS	`Adjacent-numbered pages: BP1118 BP1119 BP1120 BP1121 BP1122 * BP1124 BP1125 BP1126 BP1127 BP1128`
	EXAMPLE	`EX9124 shows a 9 square by 9 square grid. Take each tile to be 3 squares by 3 squares; there is a 3 tile by 3 tile checkerboard pattern. (One of these tiles is itself a checkerboard pattern; the other is all black squares.)`
	KEYWORD	`hard, nice, precise, allsorted, hardsort, grid, miniworlds`
	AUTHOR	`Jago Collins`

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