The On-Line Encyclopedia of Bongard Problems

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BP61

A line separates the crosses in half vs. a line does not separate the crosses in half.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 234.`
	CROSSREFS	`Adjacent-numbered pages: BP56 BP57 BP58 BP59 BP60 * BP62 BP63 BP64 BP65 BP66`
	KEYWORD	`finished, traditional, bongard`
	CONCEPT	`sides_of_line (info \| search),` `shape_cluster (info \| search),` `cluster (info \| search),` `same_number (info \| search),` `same (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

BP81

Dark figures can be divided from outline figures by a straight line vs. convex hulls of filled and outlined figures overlap.

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	REFERENCE	`M. M. Bongard, Pattern Recognition, Spartan Books, 1970, p. 240.`
	CROSSREFS	`Adjacent-numbered pages: BP76 BP77 BP78 BP79 BP80 * BP82 BP83 BP84 BP85 BP86`
	KEYWORD	`finished, traditional, bongard`
	CONCEPT	`convex_hull (info \| search),` `separable_entangled (info \| search),` `sides_of_line (info \| search),` `feature_cluster (info \| search),` `cluster_of_one (info \| search),` `cluster (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search),` `objects_overlap (info \| search),` `overlap (info \| search)`
	AUTHOR	`Mikhail M. Bongard`

BP121

Objects below match their "symbols" above, according to one "code" versus objects below match their "symbols" above, according to another "code."

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	COMMENTS	`Circle: VV; square: VΛ; triangle: ΛΛ ; blank: ΛV vs. circle: ΛΛ; square: ΛV; triangle: VV; blank: VΛ.`
	CROSSREFS	`Adjacent-numbered pages: BP116 BP117 BP118 BP119 BP120 * BP122 BP123 BP124 BP125 BP126`
	KEYWORD	`arbitrary, consistentsymbols, traditional`
	CONCEPT	`absence_as_presence (info \| search),` `correspondence (info \| search),` `sides_of_line (info \| search),` `triangle (info \| search)`
	AUTHOR	`Douglas R. Hofstadter`

BP183

Same curvature close to the middle vs. change of curvature close to the middle.

(edit; present; nest [left/right]; search; history)

	CROSSREFS	`Adjacent-numbered pages: BP178 BP179 BP180 BP181 BP182 * BP184 BP185 BP186 BP187 BP188`
	KEYWORD	`noisy, traditional`
	CONCEPT	`sides_of_line (info \| search),` `tangent (info \| search),` `imagined_point (info \| search),` `imagined_line_or_curve (info \| search),` `imagined_entity (info \| search)`
	WORLD	`curve [smaller \| same \| bigger]`
	AUTHOR	`Harry E. Foundalis`

BP1293

Line segment separates the dots into two consecutive Fibonacci numbers vs. not so.

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	COMMENTS	All examples show a single straight line segment and some dots, with no dot lying on the line. In left examples the line depicts the Fibonacci recurrence: the two groups of dots are consecutive Fibonacci numbers, and their total is the next Fibonacci number. Here 0 counts as a Fibonacci number (0, 1, 1, 2, 3, 5, ...), so a lone dot with the line beside it fits left as the split 0\|1. The six original left examples give each of the splits 0\|1, 1\|1, 1\|2, 2\|3, 3\|5, and 5\|8 exactly once, so their totals run through 1, 2, 3, 5, 8, 13. Several right examples are near misses: 2\|5 uses two Fibonacci numbers that are not adjacent in the sequence, 3\|3 repeats a term instead of pairing neighbors, and 0\|3 has a Fibonacci total but the wrong split. A dot lying exactly on the dividing line would be ambiguous, so such dots are excluded. This problem was created by hand as a test for AI models that could not have previously seen it in training; see the reference for details.
	REFERENCE	`M. Hodges, Bongard Problems, matthodges.com, 19 Aug 2026. https://matthodges.com/posts/2026-08-19-bongard-problems/`
	CROSSREFS	`Compare BP334 (even number of dots vs. odd number of dots) and BP202 (even number of shapes vs. odd number of shapes), other Problems solved by counting rather than by geometry.` `Adjacent-numbered pages: BP1288 BP1289 BP1290 BP1291 BP1292 * BP1294 BP1295`
	EXAMPLE	`The box with five dots on one side of the line and eight on the other fits on the left because 5 and 8 are consecutive Fibonacci numbers, with 13, their total, the next one. The box with two dots on one side and five on the other fits on the right because 2 and 5, although both Fibonacci numbers, are not consecutive. The box with three dots all on one side fits on the right because its split is 0\|3 rather than the 1\|2 a Fibonacci total of 3 would need.`
	KEYWORD	`precise, notso, number, math, left-narrow, traditional`
	CONCEPT	`addition (info \| search),` `number (info \| search),` `dot (info \| search),` `sides_of_line (info \| search),` `sequence (info \| search)`
	AUTHOR	`Matt Hodges`

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