The On-Line Encyclopedia of Bongard Problems

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Horizontal axis of symmetry vs. no horizontal axis of symmetry.

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	CROSSREFS	`BP1206 is the same solution but using the vertical axis instead of the horizontal axis.` `BP1215 is the same solution but with the NW/SE diagonal instead of the vertical axis.` `Adjacent-numbered pages: BP1202 BP1203 BP1204 BP1205 BP1206 * BP1208 BP1209 BP1210 BP1211 BP1212`
	KEYWORD	`stub, notso, stretch, left-narrow, traditional`
	CONCEPT	`horizontal (info \| search),` `symmetry_axis (info \| search),` `symmetry (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP559

Cross section of a cube vs. not cross section of a cube

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	COMMENTS	`All examples are solid black shapes.` `This problem is absurdly hard. It makes a good extreme example. - Aaron David Fairbanks, Nov 23 2020`
	CROSSREFS	`Adjacent-numbered pages: BP554 BP555 BP556 BP557 BP558 * BP560 BP561 BP562 BP563 BP564`
	KEYWORD	`hard, precise, allsorted, notso, stretch, challenge, left-narrow, perfect`
	CONCEPT	`cube (info \| search),` `cross_section (info \| search)`
	WORLD	`fill_shape [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP247

Five straight or curved lines vs. not five straight or curved lines.

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	CROSSREFS	`Adjacent-numbered pages: BP242 BP243 BP244 BP245 BP246 * BP248 BP249 BP250 BP251 BP252`
	KEYWORD	`traditional`
	CONCEPT	`curved_straight (info \| search),` `number (info \| search)`
	AUTHOR	`Andreas Gunnarsson`

BP850

Shape can be maneuvered around the corner vs. not so.

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	REFERENCE	`https://en.wikipedia.org/wiki/Moving_sofa_problem`
	CROSSREFS	`Adjacent-numbered pages: BP845 BP846 BP847 BP848 BP849 * BP851 BP852 BP853 BP854 BP855`
	KEYWORD	`nice, precise, physics, creativeexamples, proofsrequired, left-narrow, right-narrow, dithering`
	CONCEPT	`rotation_required (info \| search),` `imagined_motion (info \| search),` `physically_fitting (info \| search)`
	AUTHOR	`Leo Crabbe`

BP925

The numbers of dots differ by three vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP920 BP921 BP922 BP923 BP924 * BP926 BP927 BP928 BP929 BP930`
	KEYWORD	`math, unorderedpair, traditional`
	CONCEPT	`subtraction (info \| search),` `number (info \| search),` `dot (info \| search),` `three (info \| search)`
	WORLD	`two_dot_clusters [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP955

Images of Bongard Problems that sort an image of their left side on their left and an image of their right side on their left vs. images of Bongard Problems that sort an image of their left side on their right and an image of their right side on their right.

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	CROSSREFS	`See also BP957 for the other two evident possibilities.` `Adjacent-numbered pages: BP950 BP951 BP952 BP953 BP954 * BP956 BP957 BP958 BP959 BP960`
	KEYWORD	`abstract, dual, handed, leftright, solved, meta (see left/right), miniproblems, creativeexamples, assumesfamiliarity, structure, experimental`
	CONCEPT	`self-reference (info \| search)`
	WORLD	`oblong_boxes_bpimage_sorts_both_sides_skewed [smaller \| same \| bigger] zoom in left (oblong_boxes_bpimage_sorts_both_sides_left) \| zoom in right (oblong_boxes_bpimage_sorts_both_sides_right)`
	AUTHOR	`Leo Crabbe`

BP924

Polygons where all sides are different lengths vs. Polygons where not all sides are different lengths.

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	COMMENTS	`All examples in this Problem are outlines of convex polygons.` `This is a generalisation of scalene triangles to any polygon.`
	CROSSREFS	`The left side implies the right side of BP329 (regular vs. irregular polygons), but the converse is not true.` `The left side of BP329 implies the right side, but the converse is not true.` `Adjacent-numbered pages: BP919 BP920 BP921 BP922 BP923 * BP925 BP926 BP927 BP928 BP929`
	EXAMPLE	`Any scalene triangle will fit on the left, because no two sides are equal.` `However, any regular polygon will not fit on the left, because all of its sides are equal.` `A random convex polygon will "almost surely" fit on the left.`
	KEYWORD	`nice, stretch, right-narrow, traditional`
	CONCEPT	`all (info \| search)`
	WORLD	`polygon_outline [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP49

Points inside the figure outline are grouped more densely than outside the contour vs. points outside the figure contour are grouped more densely than inside the contour.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 230.`
	CROSSREFS	`Adjacent-numbered pages: BP44 BP45 BP46 BP47 BP48 * BP50 BP51 BP52 BP53 BP54`
	KEYWORD	`finished, traditional, viceversa, bongard`
	CONCEPT	`inside (info \| search),` `interior_exterior (info \| search),` `near_far (info \| search),` `feature_cluster (info \| search),` `shape_cluster (info \| search),` `cluster (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

BP1123

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`
	CONCEPT	`element_grouping (info \| search)`
	AUTHOR	`Jago Collins`

BP150

Odd number of squares vs. even number of squares.

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	CROSSREFS	`Adjacent-numbered pages: BP145 BP146 BP147 BP148 BP149 * BP151 BP152 BP153 BP154 BP155`
	KEYWORD	`number, traditional`
	CONCEPT	`even_odd (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

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