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BP824 Objects shown chosen from collection in an ordered, algorithmic way vs. random choices involved.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP819 BP820 BP821 BP822 BP823  *  BP825 BP826 BP827 BP828 BP829

KEYWORD

abstract

AUTHOR

Aaron David Fairbanks

BP825 Ticks mark an infinite sequence of angles on circle such that each angle is the double of the subsequent angle in the sequence (angle measured from rightmost indicated point) vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

This is solvable; it was solved by Sridhar Ramesh.


A full turn is considered "the same angle" as no turns; likewise for adding and subtracting full turns from any angle. All sequences of angles shown start at the rightmost tick.


It doesn't matter whether the angle is measured clockwise or counterclockwise, as long as the choice is consistent.

CROSSREFS

Adjacent-numbered pages:
BP820 BP821 BP822 BP823 BP824  *  BP826 BP827 BP828 BP829 BP830

KEYWORD

hard, convoluted, notso, math, solved

CONCEPT sequence (info | search)

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

BP1260 is about applying the same transformation to different objects.

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),
function (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

BP841 Any relationship that exists between one object and another exists between each object and some other versus not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

For example, in a picture on the left of this Bongard Problem, if object A turned 90 degrees clockwise is object B, then there is also an object C which is B turned 90 degrees clockwise.


Positioning is irrelevant.


In all images, any pair of objects ought to be related in a unique (most intuitive) way. Furthermore, one object is not allowed to be related to two distinct objects by the same relationship. Even for images on the right, each analogy of objects A:B::C:_ should have one clear answer, although that object is perhaps missing.


Relationships described by "[undo-able action] applied to ___ is ___" will always form what in mathematics is called a "group". These relationships can be chained one after another to form a total relationship (turn 90 degrees clockwise + turn 90 degrees clockwise = turn 180 degrees), and each relationship has an "inverse" relationship that undoes it and vice versa (turn 90 degrees clockwise + turn 90 degrees counterclockwise = do nothing).

(Moreover actions are by nature associative.)


Sometimes the relationships in a picture wouldn't be consistently read the same way by everybody. For example, if there is a picture showing an L shape next to all vertical and horizontal reflections and 90 degree rotations of it, somebody might read

⅃ L

to be the same relationship as

┗━

┏━.

Meanwhile, someone else might think ⅃ L should be called the same relationship as ┗━ ━┛. There is a conflict between "flipping over the vertical line within the letter 'L'" and "flipping over a vertical line in the background space."


Likewise in any illustration of related objects (as in this Bongard Problem) people might interpret [the transformation that sends A to B] as analogous to [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].


A "commutative" (also called "abelian") group is a group in which there is no difference between the two in each case. Displayed using pictures like the ones in this Bongard Problem, only commutative groups of relationships can be expected to be read consistently by people.

REFERENCE

https://en.wikipedia.org/wiki/Group_(mathematics)

https://en.wikipedia.org/wiki/Abelian_group

CROSSREFS

See BP842 and BP840 for versions about particular groups.

Adjacent-numbered pages:
BP836 BP837 BP838 BP839 BP840  *  BP842 BP843 BP844 BP845 BP846

KEYWORD

nice, rules, miniworlds

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

BP843 Lower bound is object vs. lower bound is nothing.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP838 BP839 BP840 BP841 BP842  *  BP844 BP845 BP846 BP847 BP848

KEYWORD

creativeexamples, structure, rules

CONCEPT empty (info | search)

WORLD

constant_change_seq_increase_right_lower_bound [smaller | same | bigger]
zoom in left (constant_change_seq_increase_right_low_bound_obj)

AUTHOR

Aaron David Fairbanks

BP844 Cars vs. no cars
(edit; present; nest [left/right]; search; history)
COMMENTS

Based on Google reCaptcha human verification checks.

CROSSREFS

Adjacent-numbered pages:
BP839 BP840 BP841 BP842 BP843  *  BP845 BP846 BP847 BP848 BP849

KEYWORD

nice, teach, anticomputer, culture, color, experimental, funny

WORLD

color_street_photograph [smaller | same | bigger]

AUTHOR

Cameron Fetter

BP845 "Noisy" properties changing independent of the consistently increasing quantity vs. not so.
(edit; present; nest [left/right]; search; history)
COMMENTS

"Noisy" sequences of change vs. "minimalist" sequences of change.

CROSSREFS

See BP511 for noise in BPs.

See BP827 for noise in images of BPs.

Adjacent-numbered pages:
BP840 BP841 BP842 BP843 BP844  *  BP846 BP847 BP848 BP849 BP850

KEYWORD

creativeexamples, structure, contributepairs, rules

WORLD

quantity_change_seq_increase_right_noise_allowed [smaller | same | bigger]
zoom in right (quantity_change_seq_increase_right)

AUTHOR

Aaron David Fairbanks

BP846 A quantity increases by fixed constant amount each step vs. not so.
(edit; present; nest [left/right]; search; history)
CROSSREFS

Adjacent-numbered pages:
BP841 BP842 BP843 BP844 BP845  *  BP847 BP848 BP849 BP850 BP851

KEYWORD

creativeexamples, structure, contributepairs, rules

WORLD

quantity_change_seq_increase_right [smaller | same | bigger]
zoom in right (constant_change_seq_increase_right)

AUTHOR

Aaron David Fairbanks

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