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" 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.

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" 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.

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" group is a group in which there is no difference between the two cases. 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.

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 an L shape with vertical and horizontal reflections and 90 degree rotations, 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] to be the same relationship as [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].

A "commutative" group is a group in which there is no difference between the two cases. 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.

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 we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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] to be the same relationship as [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].

A "commutative" group is a group in which there is no difference between the two cases. 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.

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 the same by everybody. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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] to be the same relationship as [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].

A "commutative" group is a group in which there is no difference between the two cases. 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.

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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] to be the same relationship as [the transformation that sends [transformation x applied to A] to [transformation x applied to B] ].

A "commutative" group is a group in which there is no difference between the two cases. 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.

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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)."

When a group is illustrated as in this Bongard Problem, and there is a difference between [the transformation that sends A to B] and [the transformation that sends [transformation 1 applied to A] to [transformation 1 applied to B]], somebody might think it is more natural to instead call these the same transformation (as in the example above).

A "commutative" group is a group in which there is no difference between the two cases. Illustrated as in this Bongard Problem, 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.

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

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

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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)."

The problem is in deciding what it should mean to apply the ⅃ --> L transformation to other objects, e.g. the object┗━.

A "commutative" group is a group in which there is no difference between [transformation 1 with transformation 2 applied before it] and [transformation 1 with transformation 2 applied after it]. When groups are illustrated as in this Bongard Problem, 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.

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

⅃

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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 world)."

A "commutative" group is a group in which there is no difference between [transformation 1 with transformation 2 applied before it] and [transformation 1 with transformation 2 applied after it]. When groups are illustrated as in this Bongard Problem, 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.

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

⅃

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 by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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 world)."

A "commutative" group is a group in which there is no difference between [transformation 1 with transformation 2 applied before it] and [transformation 1 with transformation 2 applied after it]. When groups are illustrated as in this Bongard Problem, 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.

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 won't be consistently parsed by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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 world)."

A "commutative" group is a group in which there is no difference between [transformation 1 with transformation 2 applied before it] and [transformation 1 with transformation 2 applied after it]. When groups are illustrated as in this Bongard Problem, 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.

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 nothing changed in the end).

Sometimes the relationships won't be consistently parsed by everybody in the same way. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, 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 world)."

A "commutative" group is a group in which there is no difference between [transformation 1 with transformation 2 applied before it] and [transformation 1 with transformation 2 applied after it]. When groups are illustrated as in this Bongard Problem, 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.

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 nothing changed in the end).

Relationships that will be intuitively and consistently parsed by most people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific way of parsing the relationships.

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

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.

Note relationships between objects can be chained. For example, if A turned 90 degrees clockwise is B, and B turned 90 degrees clockwise is C, then A turned 90 degrees then turned 90 degrees is C. Also note that if A is related to B in some way then B is related to A in some way, and the two relationships undo one another. The relationships form a mathematical group.

Drawings of collections fitting on the left (with a unique "relationship" between each two things) with relationships that will be intuitively and consistently parsed by all people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific 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 boxes on the left, if A turned 90 degrees clockwise is B, then there is an object C which is B turned 90 degrees clockwise.

Note relationships between objects can be chained. For example, if A turned 90 degrees clockwise is B, and B turned 90 degrees clockwise is C, then A turned 90 degrees then turned 90 degrees is C. Also note that if A is related to B in some way then B is related to A in some way, and the two relationships undo one another. The relationships form a mathematical group.

Drawings of collections fitting on the left (with a unique "relationship" between each two things) with relationships that will be intuitively and consistently parsed by all people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific 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 such relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way.

For example, in boxes on the left, if A turned 90 degrees clockwise is B, then there is an object C which is B turned 90 degrees clockwise.

Note relationships between objects can be chained. For example, if A turned 90 degrees clockwise is B, and B turned 90 degrees clockwise is C, then A turned 90 degrees then turned 90 degrees is C. Also note that if A is related to B in some way then B is related to A in some way, and the two relationships undo one another. The relationships form a mathematical group.

Drawings of collections fitting on the left (with a unique "relationship" between each two things) with relationships that will be intuitively and consistently parsed by all people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific 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 such relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way.

For example, in boxes on the left, if A turned 90 degrees clockwise is B, then there is an object C which is B turned 90 degrees clockwise.

Note relationships between objects can be chained. For example, if A turned 90 degrees clockwise is B, and B turned 90 degrees clockwise is C, then A turned 90 degrees then turned 90 degrees is C. Also note that if A is related to B in some way then B is related to A in some way, and the two relationships undo one another. The relationships form a mathematical group.

Drawings of collections fitting on the left (with a unique "relationship" between each two things) with relationships that will be intuitively and consistently parsed by people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific 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 such relationships. In all examples, any pair of objects are related to one another in a unique (most intuitive) way.

For example, in boxes on the left, if A turned 90 degrees clockwise is B, then there is an object C which is B turned 90 degrees clockwise.

Note relationships between objects can be chained. For example, if A turned 90 degrees clockwise is B, and B turned 90 degrees clockwise is C, then A turned 90 degrees then turned 90 degrees is C. Also note that if A is related to B in some way then B is related to A in some way, and the two relationships undo one another. The analogies form a mathematical group.

Drawings of collections fitting on the left (with a unique "relationship" between each two things) with relationships that will be intuitively and consistently parsed by people in the same way are generally only possible when the relationships form an "abelian group," i.e. whenever it is true that A - - -> B, B ~~> C, and A ~~> X, it is also true that X - - -> C. (A, B, C, and X are things; - - -> and ~~> are relationships.) This is to say - - - > ~~> = ~~> - - - > for any two relationships - - -> and ~~~> that things can have. For example, rotating 10 degrees clockwise then rotating 20 degrees clockwise is the same as rotating 20 degrees then 10 degrees. However, this is not true about some relationships; rotating 10 degrees clockwise then mirroring horizontally is not the same as mirroring horizontally then rotating 10 degrees clockwise. This introduces a problem when we want to represent these relationships in this Problem. If there is one interpretation of relationships - - - >, ~~~>, and = = = > where

A - - - > B

~         ~

~         ~

V         V

X = = = > C

for things A, B, C, and X, then it is likely some people might most naturally parse - - - > and = = = > as the same relationship. For example, if we have an L shape with vertical and horizontal reflections and 90 degree rotations, somebody might parse

L⅃

to be the same relationship as

┏━

┗━,

because that is L⅃ post-rotation. However, thinking in terms of performing reflections and rotations with respect to the bounding box, L⅃ is instead the same relationship as ┏━ ━┓. There will be a similar problem whenever there is a conflict between a relationship and what that relationship becomes post-[transformation]. In this example there is a conflict between "flipping with respect to the bounding box" and "flipping with respect to how L is rotated."

Sending relationships to their versions in the post-[transformation] world is more commonly called "conjugation by [transformation]." The post-[transformation] version of anything is itself for all transformations if and only if we have an abelian group.

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

See BP842 and BP840 for versions with non-abelian groups; these Problems sidestep the problem because the solutions involve a specific way of parsing the relationships.