The On-Line Encyclopedia of Bongard Problems

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BP853

Prime knot vs. composite knot.

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

	REFERENCE	`https://en.wikipedia.org/wiki/Prime_knot`
	CROSSREFS	`Adjacent-numbered pages: BP848 BP849 BP850 BP851 BP852 * BP854 BP855 BP856 BP857 BP858`
	KEYWORD	`math, hardsort`
	CONCEPT	`knot (info \| search)`
	WORLD	`knot [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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`

BP898

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

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

	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`

BP899

Regions in drawing (ignore background) can be coloured using three or fewer colours such that no adjacent regions are coloured the same colour vs. four colours are required.

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

	COMMENTS	`This Problem is related to the four colour theorem.`
	CROSSREFS	`Adjacent-numbered pages: BP894 BP895 BP896 BP897 BP898 * BP900 BP901 BP902 BP903 BP904`
	KEYWORD	`hard, nice, math`
	CONCEPT	`separated_regions (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

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`

BP925

The numbers of dots differ by three vs. not so.

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

	CROSSREFS	`Adjacent-numbered pages: BP920 BP921 BP922 BP923 BP924 * BP926 BP927 BP928 BP929 BP930`
	KEYWORD	`math, unorderedpair, traditional`
	CONCEPT	`subtraction (info \| search),` `number (info \| search),` `dot (info \| search),` `three (info \| search)`
	WORLD	`two_dot_clusters [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP926

Numbers of dots in ascending order from left to right vs. numbers of dots neither in ascending nor descending order from left to right.

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

	CROSSREFS	`Adjacent-numbered pages: BP921 BP922 BP923 BP924 BP925 * BP927 BP928 BP929 BP930 BP931`
	KEYWORD	`nice, math, sequence, traditional, left-listable, right-listable`
	CONCEPT	`number (info \| search),` `dot (info \| search)`
	WORLD	`dot_clusters_sequence_horizontal [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP989

Number of dots is n factorial for some n vs. not so.

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

	COMMENTS	`Zero is intentionally left out to avoid confusion (although it would fit right).`
	CROSSREFS	`Adjacent-numbered pages: BP984 BP985 BP986 BP987 BP988 * BP990 BP991 BP992 BP993 BP994`
	KEYWORD	`stub, precise, number, math, left-narrow, right-null, help, preciseworld`
	WORLD	`dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1024

One sorted left vs. one sorted right.

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

	CROSSREFS	`Adjacent-numbered pages: BP1019 BP1020 BP1021 BP1022 BP1023 * BP1025 BP1026 BP1027 BP1028 BP1029`
	KEYWORD	`precise, allsorted, dual, handed, leftright, math, meta (see left/right), miniproblems, assumesfamiliarity, structure, preciseworld, presentationinvariant`
	WORLD	`boxes_dots_bpimage_clear_set_of_numbers [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

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