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.

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.

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 compound 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.

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 "[undoable 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 compound 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.

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 "[undoable 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 compound 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. For pictures like the ones in this Bongard Problem, only commutative groups of relationships can be expected to be read consistently by people.

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 "[undoable 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 compound 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 "[undoable 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 compound 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 answer, although that object is perhaps missing.

Relationships described by "[undoable 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 compound 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships described by "[undoable 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 compound 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships described by "[undoable 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 compound 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).

(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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships described by "[undoable 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 compound 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).

(And 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships described by "[undoable 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 compound 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). (Also, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships like these--"[undoable action] applied to __is__"--will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another 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 answer, although that object is perhaps missing.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another in a unique (most intuitive) way. Furthermore, one object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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 to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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.

Examples of relationships: "__turned 90 degrees is ___" and "___ flipped horizontally is__". In all images, any pair of objects ought to be related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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. For pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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.

Examples of relationships: "__turned 90 degrees is ___" and "___ flipped horizontally is__". In all images, any pair of objects ought to be related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 naturally 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 pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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.

Examples of relationships: "__turned 90 degrees is ___" and "___ flipped horizontally is__". In all images, any pair of objects ought to be related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

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.

Examples of relationships: "__turned 90 degrees is ___" and "___ flipped horizontally is__". In all images, any pair of objects ought to be related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

Relationships like these ("[undoable action] applied to __is__") will always form what in mathematics is called a "group". Any relationships can be chained one after another to form a total compound 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). (Furthermore, 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 and 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

Positioning is irrelevant.

"__turned 90 degrees is ___" and "___ flipped horizontally is__" are examples of relationships. In all images, any pair of objects are related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

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.

In all the examples of this Bongard Problem, the relationships form what in mathematics is called a "group". The relationships can be chained one after another to form other relationships (example: "turned 90 degrees clockwise then turned 30 degrees clockwise" is its own relationship, "turned 120 degrees clockwise"), and each relationship has an "inverse" relationship that undoes it (example: "turned 90 degrees clockwise then turned 90 degrees counterclockwise" and vice versa both mean in the end nothing has changed).

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 and 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

See BP842 and BP840 for versions about particular groups.

Positioning is irrelevant.

"__turned 90 degrees is ___" and "___ flipped horizontally is__" are examples of relationships. In all images, any pair of objects are related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

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.

In all the examples of this Bongard Problem, the relationships form what in mathematics is called a "group". The relationships can be chained one after another to form other relationships (example: "turned 90 degrees clockwise then turned 30 degrees clockwise" is its own relationship, "turned 120 degrees clockwise"), and each relationship has an "inverse" relationship that undoes it (example: "turned 90 degrees clockwise then turned 90 degrees counterclockwise" and vice versa both mean in the end nothing has changed).

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 and 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 pictures like those in this Bongard Problem, only commutative groups of relationships will be read consistently by people.

See BP842 and BP840 for versions with non-commutative groups. These Bongard Problems sidestep the problem because the solutions specify a particular way of reading the relationships.

Positioning is irrelevant.

"__turned 90 degrees is ___" and "___ flipped horizontally is__" are some relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way. One object is not allowed to be related to two distinct objects by the same relationship.

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.

In all the examples of this Bongard Problem, the relationships form what in mathematics is called a "group". The relationships can be chained one after another to form other relationships (example: "turned 90 degrees clockwise then turned 30 degrees clockwise" is its own relationship, "turned 120 degrees clockwise"), and each relationship has an "inverse" relationship that undoes it (example: "turned 90 degrees clockwise then turned 90 degrees counterclockwise" and vice versa both mean in the end nothing has changed).

Sometimes the relationships in a picture won't be consistently read one way by everybody. For example, if there is a picture showing an L shape and 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 never difference between the two. Only commutative groups will be read consistently.

We could decide to include non-commutative groups (and other drawings in which there is more than one choice of relationships) in this Bongard Problem as long as regardless of which relationships were seen the example would be sorted by the Bongard Problem the same way. We could also instead add a caveat to the solution of this Bongard Problem "there exists a choice of relationships such that..." However, these decisions may cloud the idea.

See BP842 and BP840 for versions with non-commutative groups. These Bongard Problems sidestep the problem because the solutions specify a particular way of parsing the relationships.

Positioning is irrelevant.

Here "relationship" is meant in the sense that one object is not allowed to be related to two distinct objects by the same relationship. "... turned 90 degrees is ..." and "... flipped horizontally is ..." are some relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way.

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.

In all the examples of this Bongard Problem, the relationships form what in mathematics is called a "group". The relationships can be chained one after another to form other relationships (example: "turned 90 degrees clockwise then turned 30 degrees clockwise" is its own relationship, "turned 120 degrees clockwise"), and each relationship has an "inverse" relationship that undoes it (example: "turned 90 degrees clockwise then turned 90 degrees counterclockwise" and vice versa both mean in the end nothing has changed).

Sometimes the relationships in a picture won't be consistently read one way by everybody. For example, if there is a picture showing an L shape and 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 never difference between the two. Only commutative groups will be read consistently.

We could decide to include non-commutative groups (and other drawings in which there is more than one choice of relationships) in this Bongard Problem as long as regardless of which relationships were seen the example would be sorted by the Bongard Problem the same way. We could also instead add a caveat to the solution of this Bongard Problem "there exists a choice of relationships such that..." However, these decisions may cloud the idea.

See BP842 and BP840 for versions with non-commutative groups. These Bongard Problems sidestep the problem because the solutions specify a particular way of parsing the relationships.

Positioning is irrelevant.

Here "relationship" is meant in the sense that one object is not allowed to be related to two distinct objects by the same relationship. "... turned 90 degrees is ..." and "... flipped horizontally is ..." are some relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way.

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.

In all the examples of this Bongard Problem, the relationships form what in mathematics is called a "group". The relationships can be chained one after another to form other relationships (example: "turned 90 degrees clockwise then turned 30 degrees clockwise" is its own relationship, "turned 120 degrees clockwise"), and each relationship has an "inverse" relationship that undoes it (example: "turned 90 degrees clockwise then turned 90 degrees counterclockwise" and vice versa both mean in the end nothing has changed).

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

⅃ 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 the vertical line (in the background space)."

Likewise in any illustration of related objects (as in this Bongard Problem) people might naturally 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 never difference between the two. Only commutative groups will be parsed consistently.

We could decide to include non-commutative groups (and other drawings in which there is more than one choice of relationships) in this Bongard Problem as long as regardless of which relationships were seen the example would be sorted by the Bongard Problem the same way. We could also instead add a caveat to the solution of this Bongard Problem "there exists a choice of relationships such that..." However, these decisions may cloud the idea.

See BP842 and BP840 for versions with non-commutative groups. These Bongard Problems sidestep the problem because the solutions specify a particular way of parsing the relationships.