To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row." (Even just a random choice of squares is probably more intuitive than that, so maybe it is impossible for that to be the most intuitive rule.)

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row." (Even just a random choice of squares is probably more intuitive than that, so maybe it is impossible for that to be the most intuitive rule.)

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row." (Even just a random choice of squares is probably more intuitive than that, so maybe it's impossible for that to be the most intuitive rule.)

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row." (Even just a random choice of squares is probably more intuitive than that, so maybe it's impossible for it to be the most intuitive rule.)

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row." (Even just a random choice of squares is probably more intuitive than that, so maybe it's impossible for this to be the most intuitive rule.)

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to work with once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution with an example: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but it introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" are assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with sixteen dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem should show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It introduces a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. It shows a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. Still, it shows a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve. Still, this Problem shows a structure (see BP789) that might lead to more interesting Problems.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

Although this Problem's solution seems somewhat convoluted, this structure turns out to be reasonably intuitive to make Bongard Problems about once it is defined, so this is a "teach" Problem (left-BP858); with many more examples it could be used to nonverbally teach this structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This Problem's solution seems somewhat convoluted at first, but, once defined, this structure turns out to be reasonably intuitive to make Bongard Problems about, so this is a "teach" Problem (left-BP858); it could be used to nonverbally teach this structure to somebody, similarly to BP968. (See BP789 for more examples of structures like that.)

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be parsed so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This Problem's solution seems somewhat convoluted at first, but, once defined, this structure turns out to be reasonably intuitive to make Bongard Problems about, so this is a "teach" Problem (left-BP858); it could be used to nonverbally teach this structure to somebody, similarly to BP968. (See BP789 for more examples of structures like that.)

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be thought of so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

This Problem's solution seems somewhat convoluted at first, but, once defined, this structure turns out to be reasonably intuitive to make Bongard Problems about, so this is a "teach"ing Problem (left-BP858); it could be used to nonverbally teach this structure to somebody, similarly to BP968. (See BP789 for more examples of structures like that.)

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be thought of so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution with more mathematical jargon is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so." Another equivalent solution is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

An expanded version of this Bongard Problem could be used to nonverbally teach this particular generalized analogy structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be thought of so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

To clarify the solution: on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution worded slightly more mathematically is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Another equivalent solution is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so."

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

An expanded version of this Bongard Problem could be used to nonverbally teach this particular generalized analogy structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be thought of so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

To each column is assigned something; to each row is assigned something; columns and rows combine in images via a consistent rule vs. not so.

This Bongard Problem may not be very satisfying to solve, but there are some interesting examples.

For example, on the left is an image of a grid where the first row features a square with three dots and a square with nine dots, and the second row features a square with four dots and square with nine dots. "Three" and "four" assigned to the rows; "x" and "x squared" are assigned to the columns.

An equivalent solution worded slightly more mathematically is "columns (or rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

Another equivalent solution is "defines an intuitive map from the cartesian product of two sets to a third set vs. not so."

Left examples are a generalized version of the analogy structures seen in BP361. Any typical visual analogy a : b :: c : d shown in a 2x2 grid will fit on the left of this Problem.

An expanded version of this Bongard Problem could be used to nonverbally teach this particular generalized analogy structure to somebody, similarly to BP968.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey. The "rule" can be about how the images relate to their neighbors, it can involve the position of the images in the grid, and it can involve properties of the grid considered as a whole.

There is a trivial way in which any example can be thought of so that it fits on the left side: imagine each column is assigned the list of all the squares in that column and each row is assigned the list of all the squares in that row. Then the rows and columns will indeed combine in a natural way to give the images shown. As such, the solution of this Problem assumes use of more natural interpretations of rules for these grids. All grids in this Problem show a rule that is more intuitive to parse than "assign each column the list of all the squares in it and each row the list of all the squares in it and let a square in the grid show the respective square assigned at that place in the corresponding column and row."

See BP979 for use of similar structures but with one square removed from the grid.

Aaron David Fairbanks