The On-Line Encyclopedia of Bongard Problems

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Two clusters of 2+3, middle shape must be seen as rectangle (AND only one linelike shape) vs. two clusters of 2+3, middle shape must be seen as piece of straight line (AND more than one linelike shape).

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	CROSSREFS	`Adjacent-numbered pages: BP154 BP155 BP156 BP157 BP158 * BP160 BP161 BP162 BP163 BP164`
	KEYWORD	`convoluted, unwordable`
	CONCEPT	`number (info \| search)`
	AUTHOR	`Harry E. Foundalis`

BP1066

Some region has one box in it vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP1061 BP1062 BP1063 BP1064 BP1065 * BP1067 BP1068 BP1069 BP1070 BP1071`
	KEYWORD	`perfect, infinitedetail`
	CONCEPT	`fractal (info \| search),` `recursion (info \| search),` `self-reference (info \| search),` `one (info \| search)`
	WORLD	`recursive_boxes [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP137

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`

BP144

Two clusters of three and two: different features (including clustering) yield different 3-2 splits vs. two clusters of three and two: different features yield the same 3-2 split.

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	CROSSREFS	`Adjacent-numbered pages: BP139 BP140 BP141 BP142 BP143 * BP145 BP146 BP147 BP148 BP149`
	KEYWORD	`nice, traditional, miniworlds`
	CONCEPT	`and (info \| search),` `number (info \| search),` `separable_entangled (info \| search),` `feature_cluster (info \| search),` `shape_cluster (info \| search),` `cluster (info \| search),` `two (info \| search),` `three (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

BP360

Order is high (low entropy) vs. order is low (high entropy).

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	COMMENTS	`"Order versus chaos."`
	CROSSREFS	`Adjacent-numbered pages: BP355 BP356 BP357 BP358 BP359 * BP361 BP362 BP363 BP364 BP365`
	KEYWORD	`nice, abstract, collective, contributepairs, traditional, miniworlds`
	CONCEPT	`pattern_or_random (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP192

Elongated vertically vs. elongated horizontally.

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	CROSSREFS	`Adjacent-numbered pages: BP187 BP188 BP189 BP190 BP191 * BP193 BP194 BP195 BP196 BP197`
	KEYWORD	`nice, noisy, traditional`
	CONCEPT	`convex_hull (info \| search),` `elongated_compact (info \| search),` `horizontal (info \| search),` `vertical (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Harry E. Foundalis`

BP1050

One color is multiple of the other vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP1045 BP1046 BP1047 BP1048 BP1049 * BP1051 BP1052 BP1053 BP1054 BP1055`
	KEYWORD	`precise, allsorted, preciseworld`
	CONCEPT	`ratio (info \| search)`
	WORLD	`outline_or_fill_circles_same_size [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP981

Grid of analogies vs. different kind of rule.

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	COMMENTS	`On the left, each row and column could be labeled by a certain object or concept; on the right this is not so.` `More specifically: on the left, each row and each column is associated with a certain object or concept; there is a rule for combining rows and columns to give images; it would be possible without changing the rule to extend with new rows/columns or delete/reorder any existing columns. On the right, this is not so; the rule might be about how the images must relate to their neighbors, for example.` `All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey.` `Left examples are a generalized version of the analogy grids seen in BP361. Any analogy a : b :: c : d shown in a 2x2 grid will fit on the left here.` `To word the solution with mathematical jargon, "defines a (simply described) map from the Cartesian product of two sets vs. not so." Another equivalent solution is "columns (alternatively, rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.` `There is a trivial way in which any example can be interpreted so that it fits on the left side: imagine each row is assigned the list of all the squares in that row and each column is assigned its number, counting from the left. But each grid has a clear rule that is simpler than this.`
	CROSSREFS	`BP1258 is a similar idea: "any square removed could be reconstructed vs. not." Examples included left here usually fit left there, but some do not e.g. EX9998.` `See BP979 for use of similar structures but with one square removed from the grid.` `Adjacent-numbered pages: BP976 BP977 BP978 BP979 BP980 * BP982 BP983 BP984 BP985 BP986`
	KEYWORD	`nice, convoluted, unwordable, notso, teach, structure, rules, grid, miniworlds`
	CONCEPT	`analogy (info \| search)`
	WORLD	`grid_of_images_with_rule [smaller \| same \| bigger] zoom in left (grid_of_analogies)`
	AUTHOR	`Aaron David Fairbanks`

BP1207

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`

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