The On-Line Encyclopedia of Bongard Problems

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BP576

Vertices may be partitioned into two sets such that no two vertices in the same set are connected versus not so.

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

	COMMENTS	`Left examples are called "bipartite graphs."`
	CROSSREFS	`Adjacent-numbered pages: BP571 BP572 BP573 BP574 BP575 * BP577 BP578 BP579 BP580 BP581`
	KEYWORD	`precise, allsorted, notso, math, traditional, preciseworld`
	CONCEPT	`graph (info \| search),` `distinguishing_crossing_curves (info \| search)`
	WORLD	`graph [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP788

Graph contains a "loop" a.k.a. cycle (cyclic) versus graph is acyclic.

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

	CROSSREFS	`Adjacent-numbered pages: BP783 BP784 BP785 BP786 BP787 * BP789 BP790 BP791 BP792 BP793`
	KEYWORD	`nice, precise, allsorted, math, traditional, preciseworld`
	CONCEPT	`graph (info \| search),` `distinguishing_crossing_curves (info \| search),` `loop (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP820

Shape can be combined with a copy of itself to form a convex shape vs. not so.

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

	CROSSREFS	`For the generalization of this property, see BP991.` `Adjacent-numbered pages: BP815 BP816 BP817 BP818 BP819 * BP821 BP822 BP823 BP824 BP825`
	KEYWORD	`nice, precise, allsorted`
	CONCEPT	`tiling (info \| search)`
	WORLD	`fill_shape [smaller \| same \| bigger] zoom in left`
	AUTHOR	`Isaac Hathaway`

BP856

Compound shape would hit the dot if rotated vs. not so.

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

	CROSSREFS	`Adjacent-numbered pages: BP851 BP852 BP853 BP854 BP855 * BP857 BP858 BP859 BP860 BP861`
	KEYWORD	`nice, precise, allsorted, left-narrow, preciseworld`
	CONCEPT	`imagined_motion (info \| search),` `collision (info \| search)`
	AUTHOR	`Leo Crabbe`

BP863

Two shapes can tessellate the plane together vs. not so.

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

	CROSSREFS	`Adjacent-numbered pages: BP858 BP859 BP860 BP861 BP862 * BP864 BP865 BP866 BP867 BP868`
	KEYWORD	`nice, precise, allsorted, math, hardsort, creativeexamples, unorderedpair`
	CONCEPT	`infinite_plane (info \| search),` `tessellation (info \| search),` `tiling (info \| search)`
	WORLD	`2_shapes [smaller \| same \| bigger]`
	AUTHOR	`Leo Crabbe`

BP891

Dots can be connected to create one triangle within another vs. not so.

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	COMMENTS	`Or equivalently, "3 dots within boundary of convex hull vs. not so". - Leo Crabbe, Aug 01 2020`
	CROSSREFS	`Adjacent-numbered pages: BP886 BP887 BP888 BP889 BP890 * BP892 BP893 BP894 BP895 BP896`
	KEYWORD	`nice, precise, allsorted, preciseworld`
	CONCEPT	`convex_hull (info \| search),` `triangle (info \| search)`
	WORLD	`6_dots [smaller \| same \| bigger]`
	AUTHOR	`Cameron Fetter`

BP897

Wide angles connected to narrow angles vs. not so.

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

	COMMENTS	`Another solution is that right examples can be folded down flat onto one isosceles triangle while left examples cannot.` `All examples in this Problem feature four isosceles triangles connected by corners and/or edges.`
	CROSSREFS	`This was conceived as a false solution for BP898.` `Adjacent-numbered pages: BP892 BP893 BP894 BP895 BP896 * BP898 BP899 BP900 BP901 BP902`
	KEYWORD	`precise, allsorted, notso, traditional, preciseworld`
	CONCEPT	`triangle (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak, Aaron David Fairbanks`

BP898

Can fold into tetragonal disphenoid ("isosceles tetrahedron") vs. cannot.

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	COMMENTS	`Which two sides are the long sides and which side is the short side, or equivalently which angles are the wider angles and which angle is the narrower angle, is the only relevant information to consider for each triangle. Triangles are all assumed isosceles and congruent to one another.` `All examples in this Problem feature four of these triangles connected by corners and/or edges.`
	CROSSREFS	`BP897 was conceived as a false solution for this.` `Adjacent-numbered pages: BP893 BP894 BP895 BP896 BP897 * BP899 BP900 BP901 BP902 BP903`
	KEYWORD	`hard, precise, allsorted, notso, math, preciseworld`
	CONCEPT	`triangle (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak`

BP905

Graph can be redrawn such that no edges intersect vs. not so.

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

	COMMENTS	`A graph is a collection of vertices and edges. Vertices are the dots and edges are the lines that connect the dots. On the left, all edges can be redrawn (curved lines are allowed and moving vertices is allowed) such that no edges cross each other and each vertex is still connected to the same other vertices. These graphs are called planar.`
	CROSSREFS	`Adjacent-numbered pages: BP900 BP901 BP902 BP903 BP904 * BP906 BP907 BP908 BP909 BP910`
	KEYWORD	`nice, precise, allsorted, notso, math, left-null, preciseworld`
	CONCEPT	`graph (info \| search),` `topological_transformation (info \| search)`
	WORLD	`connected_graph [smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak`

BP922

One row is rearranged to make the other by swapping an odd number of object pairs vs. one row is rearranged to make the other by swapping an even number of object pairs.

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

	COMMENTS	`The mathematical terms for these operations are even and odd permutations.`
	CROSSREFS	`Adjacent-numbered pages: BP917 BP918 BP919 BP920 BP921 * BP923 BP924 BP925 BP926 BP927`
	KEYWORD	`precise, allsorted, math, left-narrow, right-narrow, unorderedpair, preciseworld, left-listable`
	CONCEPT	`even_odd (info \| search),` `permutation (info \| search)`
	AUTHOR	`Leo Crabbe`

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