The On-Line Encyclopedia of Bongard Problems

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BP562

There exists a closed trail that hits each vertex exactly once vs. not so.

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	COMMENTS	`Left examples are called "Hamiltonian graphs."`
	CROSSREFS	`Adjacent-numbered pages: BP557 BP558 BP559 BP560 BP561 * BP563 BP564 BP565 BP566 BP567`
	KEYWORD	`math, traditional`
	CONCEPT	`graph (info \| search),` `distinguishing_crossing_curves (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP560

There exists a closed trail that hits each edge exactly once vs. not so.

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	COMMENTS	`Left examples are called "Eulerian graphs."` `A connected graph is Eulerian if and only if each vertex is incident to an even number of edges.`
	CROSSREFS	`Adjacent-numbered pages: BP555 BP556 BP557 BP558 BP559 * BP561 BP562 BP563 BP564 BP565`
	KEYWORD	`precise, allsorted, math, traditional, preciseworld`
	CONCEPT	`graph (info \| search),` `distinguishing_crossing_curves (info \| search),` `all (info \| search),` `even_odd (info \| search),` `existence (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	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`

BP558

Point sequence that is increasing or decreasing in height vs. point sequence that alternates in height

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	COMMENTS	`All examples in this Problem are groups of 3 dots.` `Any example where 2 adjacent dots have the same height would be ambiguously sorted.`
	CROSSREFS	`Adjacent-numbered pages: BP553 BP554 BP555 BP556 BP557 * BP559 BP560 BP561 BP562 BP563`
	EXAMPLE	`Reading from right to left in the first box on the left hand side: the 2nd dot is higher than the 1st, and the 3rd is higher than the 2nd, so the sequence of dots is strictly increasing in height.`
	KEYWORD	`nice, precise, antihuman, orderedtriplet, preciseworld`
	CONCEPT	`coordinate (info \| search)`
	WORLD	`three_points [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP557

Equal horizontal length vs. not

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	COMMENTS	`All examples in this Problem are pairs of straight line segments.` `This problem communicates the idea of projected distance, in this case from 2D to 1D (x-axis).`
	CROSSREFS	`Adjacent-numbered pages: BP552 BP553 BP554 BP555 BP556 * BP558 BP559 BP560 BP561 BP562`
	KEYWORD	`nice, precise, allsorted, stretch, perfect, unorderedpair, preciseworld`
	CONCEPT	`projection (info \| search)`
	WORLD	`two_segments [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP555

Black and white photograph versus drawing.

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	COMMENTS	`Although the images here are related to culture (BP521^left), the solution of this Bongard Problem is about the visual style of the images.`
	CROSSREFS	`Adjacent-numbered pages: BP550 BP551 BP552 BP553 BP554 * BP556 BP557 BP558 BP559 BP560`
	KEYWORD	`less, culture, experimental`
	WORLD	`zoom in left (black_and_white_photograph)`
	AUTHOR	`Aaron David Fairbanks`

BP551

Unstable balance vs. not

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	REFERENCE	`https://en.wikipedia.org/wiki/Stability_theory`
	CROSSREFS	`Adjacent-numbered pages: BP546 BP547 BP548 BP549 BP550 * BP552 BP553 BP554 BP555 BP556`
	EXAMPLE	`The classic example is a pendulum with a solid rod (instead of a string) which has a stable balance point at the bottom of its swing, where if you move the pendulum slightly it will swing back towards that balanced state. However, theoretically the pendulum can also be balanced pointing directly up. In this case, if you move the pendulum slightly it will swing down away from that upwards balanced state.`
	KEYWORD	`updown, rotate, physics, anticomputer, perfect`
	CONCEPT	`tumbles_or_stays_put (info \| search),` `gravity (info \| search)`
	AUTHOR	`Jago Collins`

BP548

Due to pixelation, it is ambiguous whether the middle small shape is a resized version of the left shape or the right shape vs. there is a clear choice.

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	COMMENTS	`A strange Problem relying on pixelation of digital images.`
	CROSSREFS	`Adjacent-numbered pages: BP543 BP544 BP545 BP546 BP547 * BP549 BP550 BP551 BP552 BP553`
	KEYWORD	`less, creativeexamples, pixelperfect, structure, orderedtriplet, experimental`
	CONCEPT	`imperfection_small (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP545

Solution for this problem (BP545) in English text vs. not.

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	COMMENTS	`This Problem admits multiple solutions besides "Solution for this problem in English text vs. not so," some of which are listed as the left examples. Some do not depend on the English language, so this Problem is technically "solvable" without the relevant cultural information.`
	CROSSREFS	`Adjacent-numbered pages: BP540 BP541 BP542 BP543 BP544 * BP546 BP547 BP548 BP549 BP550`
	KEYWORD	`less, notso, culture, finished, invalid, experimental, funny`
	CONCEPT	`recursion (info \| search),` `self-reference (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP543

Image depicting infinitely many objects vs. image depicting finitely many objects.

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	COMMENTS	`Another solution is "evokes infinity vs. not so." Or "seen as having arbitrarily fine detail." All left examples show objects getting smaller and smaller and closer and closer together approaching some limit within the box.`
	CROSSREFS	`Adjacent-numbered pages: BP538 BP539 BP540 BP541 BP542 * BP544 BP545 BP546 BP547 BP548`
	KEYWORD	`nice, abstract, concept, infinitedetail`
	CONCEPT	`finite_infinite (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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