The On-Line Encyclopedia of Bongard Problems

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Figures can be transformed into one another by smooth stretching (before and after there are the same crossroad-points; there is a curve connecting points before if and only if there is a curve connecting those points after) vs. not so.

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

	COMMENTS	`Left examples are topologically homeomorphic figures.` `For some examples one can imagine pulling the shape "out of" the 2d square in 3d in order to transform it, and then laying it flat back in the 2d square. See BP810 for the version where this is not allowed.`
	CROSSREFS	`Adjacent-numbered pages: BP804 BP805 BP806 BP807 BP808 * BP810 BP811 BP812 BP813 BP814`
	KEYWORD	`nice, math, unorderedpair, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`two_figures_made_of_curves [smaller \| same \| bigger] zoom in left (two_homeomorphic_figures_made_of_curves)`
	AUTHOR	`Aaron David Fairbanks`

BP810

Figures can be transformed into one another by smooth stretching (intersection points stay constant; paths connecting those points remain), while remaining within the 2d box vs. movement out of the plane required.

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

	CROSSREFS	`All examples here fit left in BP809, a version where the figures are allowed to pass through themselves while being deformed.` `Adjacent-numbered pages: BP805 BP806 BP807 BP808 BP809 * BP811 BP812 BP813 BP814 BP815`
	KEYWORD	`nice, math, unorderedpair, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`two_homeomorphic_figures_made_of_curves [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP839

Opposite (inverse) transformations have been applied to the same specific small square on opposite sides of the dividing line versus not so.

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

	COMMENTS	`The original pre-transformed square is the same across all examples, however it is not shown in most examples; what the pre-transformed square looks like must be deduced by the solver.`
	CROSSREFS	`Adjacent-numbered pages: BP834 BP835 BP836 BP837 BP838 * BP840 BP841 BP842 BP843 BP844`
	KEYWORD	`easy, abstract, arbitrary, anticomputer, traditional, rules`
	CONCEPT	`square (info \| search),` `specificity (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP840

Any transformation (rotation or flip) that sends one L to another L sends each L to some other L versus not so.

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

	COMMENTS	`Restriction of BP841 to these axis-aligned L-shapes.` `Left examples represent subgroups of the dihedral group D4.`
	CROSSREFS	`Adjacent-numbered pages: BP835 BP836 BP837 BP838 BP839 * BP841 BP842 BP843 BP844 BP845`
	KEYWORD	`traditional`
	WORLD	`zoom in left \| zoom in right`
	AUTHOR	`Aaron David Fairbanks`

BP842

Any permutation of positions that sends one string of symbols to another sends each string of symbols to some other versus not so.

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	COMMENTS	`Restriction of BP841 to permutations.`
	CROSSREFS	`Adjacent-numbered pages: BP837 BP838 BP839 BP840 BP841 * BP843 BP844 BP845 BP846 BP847`
	KEYWORD	`hard, contributepairs, traditional`
	CONCEPT	`permutation (info \| search)`
	WORLD	`zoom in left \| zoom in right`
	AUTHOR	`Aaron David Fairbanks`

BP851

Figure with points (small white circles) can be smoothly deformed within the 2D plane without passing through itself so that all points touch to make the other figure vs. not so (movement out of the plane required).

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

	CROSSREFS	`All examples here fit left in BP369, a version where the figure is allowed to pass through itself while being deformed.` `Adjacent-numbered pages: BP846 BP847 BP848 BP849 BP850 * BP852 BP853 BP854 BP855 BP856`
	KEYWORD	`math, traditional`
	CONCEPT	`topological_transformation (info \| search)`
	WORLD	`figure_made_of_curves_and_quotient_by_hollow_dots [smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP897

Wide angles connected to narrow angles vs. not so.

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	COMMENTS	`Another way to phrase this 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	`nice, precise, allsorted, notso, traditional, preciseworld`
	CONCEPT	`triangle (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Molly C Klenzak, Aaron David Fairbanks`

BP924

Polygons where all sides are different lengths vs. Polygons where not all sides are different lengths.

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	COMMENTS	`All examples in this Problem are outlines of convex polygons.` `This is a generalisation of scalene triangles to any polygon.`
	CROSSREFS	`The left side implies the right side of BP329 (regular vs. irregular polygons), but the converse is not true.` `The left side of BP329 implies the right side, but the converse is not true.` `Adjacent-numbered pages: BP919 BP920 BP921 BP922 BP923 * BP925 BP926 BP927 BP928 BP929`
	EXAMPLE	`Any scalene triangle will fit on the left, because no two sides are equal.` `However, any regular polygon will not fit on the left, because all of its sides are equal.` `A random convex polygon will "almost surely" fit on the left.`
	KEYWORD	`nice, stretch, right-narrow, traditional`
	CONCEPT	`all (info \| search)`
	WORLD	`polygon_outline [smaller \| same \| bigger]`
	AUTHOR	`Jago Collins`

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`

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