The On-Line Encyclopedia of Bongard Problems

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BP384

Square number of dots vs. non-square number of dots.

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	COMMENTS	`All examples in this Problem are a collection of dots.` `An equivalent solution is "Dots can be arranged into a square lattice whose convex hull is a square vs. not so". - Leo Crabbe, Aug 01 2020`
	CROSSREFS	`Adjacent-numbered pages: BP379 BP380 BP381 BP382 BP383 * BP385 BP386 BP387 BP388 BP389`
	EXAMPLE	`A single dot fits because 1 = 11.` `A pair of dots does not fit because there is no integer x such that 2 = xx.`
	KEYWORD	`nice, precise, allsorted, number, math, left-narrow, left-null, help, traditional, preciseworld, collection`
	CONCEPT	`square_number (info \| search)`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP386

Lower shape can be used as a tile to build the upper one vs. not so.

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	CROSSREFS	`Adjacent-numbered pages: BP381 BP382 BP383 BP384 BP385 * BP387 BP388 BP389 BP390 BP391`
	KEYWORD	`nice, precise, allsorted, left-narrow, perfect, pixelperfect, orderedpair, traditional, preciseworld, left-listable, right-listable`
	CONCEPT	`tiling (info \| search)`
	AUTHOR	`Jago Collins`

BP389

Loops are entangled (in 3-D) vs. loops can be separated (in 3-D).

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	CROSSREFS	`Adjacent-numbered pages: BP384 BP385 BP386 BP387 BP388 * BP390 BP391 BP392 BP393 BP394`
	KEYWORD	`nice, precise, allsorted, contributepairs, traditional`
	CONCEPT	`knot (info \| search),` `topological_transformation (info \| search),` `imagined_motion (info \| search),` `motion (info \| search)`
	WORLD	`link_two_knots [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP390

Each graph vertex is uniquely defined by its connections (the graph does not admit nontrivial automorphisms) vs. the graph admits nontrivial automorphisms.

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	CROSSREFS	`Adjacent-numbered pages: BP385 BP386 BP387 BP388 BP389 * BP391 BP392 BP393 BP394 BP395`
	KEYWORD	`precise, allsorted, notso, traditional, preciseworld`
	CONCEPT	`graph (info \| search),` `self-reference (info \| search),` `topological_transformation (info \| search),` `imagined_shape (info \| search),` `imagined_entity (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

BP527

Each black filled circle belongs to exactly one large circle outline vs. not so.

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	COMMENTS	`On the left, circles define a partition of the dots.`
	CROSSREFS	`Adjacent-numbered pages: BP522 BP523 BP524 BP525 BP526 * BP528 BP529 BP530 BP531 BP532`
	KEYWORD	`precise, allsorted, traditional`
	CONCEPT	`circle (info \| search),` `exists_one (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

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`

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`

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`

BP564

Discrete points intersecting boundary of convex hull vs. connected segment intersecting boundary of convex hull

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	COMMENTS	`If a "string" is wound tightly around the shape, does one of its segments lie directly on the shape?` `All examples in this Problem are connected line segments or curves.` `We are taking lines here to be infinitely thin, so that if the boundary of the convex hull intersects the endpoint of a line exactly it is understood that they meet at 1 point.`
	CROSSREFS	`Adjacent-numbered pages: BP559 BP560 BP561 BP562 BP563 * BP565 BP566 BP567 BP568 BP569`
	EXAMPLE	`Imagine wrapping a string around the pointed star. This string would take the shape of the boundary of the star's convex hull (a regular pentagon), and would only touch the star at the end of each of its 5 individual tips, therefore the star belongs on the left.`
	KEYWORD	`hard, nice, allsorted, solved, perfect`
	AUTHOR	`Leo Crabbe`

BP569

Triangular number of dots vs. non-triangular number of dots

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	COMMENTS	`All examples in this Problem are groups of black dots.` `The nth triangular number is the sum over the natural numbers from 1 to n, where n > 0. Note: 0 is the 0th triangular number. The first few triangular numbers are 0, 1, 3 (= 1+2) and 6 (= 1+2+3)`
	CROSSREFS	`Adjacent-numbered pages: BP564 BP565 BP566 BP567 BP568 * BP570 BP571 BP572 BP573 BP574`
	KEYWORD	`nice, precise, allsorted, notso, number, math, left-narrow, left-null, help, preciseworld`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

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