Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value. One side is lesser and the other is greater.

Spectra can be either continuous or discrete.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of these is typically only possible to notice when the previous conditions are true.

If a spectrum Bongard Problem obeys 1) and 2), then it will be "exact" (left-BP508).

For example:

"Big vs. small" is not exact.

"Angles less than 90° vs. angles greater than 90°" is exact.

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will be "allsorted" (left-BP509).

For example:

"Angles less than 90° vs. angles greater than 90°" is not allsorted.

"Angles less than or equal to 90° vs. angles greater than 90°" is allsorted.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of these is typically only possible to notice when the previous conditions are true.

If a spectrum Bongard Problem obeys 1) and 2), then it will be "exact" (left-BP508).

For example:

"Big vs. small" is not exact.

"Angles less than 90° vs. angles greater than 90°" is exact.

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will be "allsorted" (left-BP509).

For example:

"Angles less than 90° vs. angles greater than 90°" is not allsorted.

"Angles less than or equal to 90° vs. angles greater than 90°" is allsorted.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of these is typically only possible to notice when the previous conditions are true.

If a spectrum Bongard Problem obeys 1) and 2), then it will be "exact" (left-BP508).

For example:

"Big vs. small" is not exact.

"Angles a lot less than about 90° vs. angles a lot greater than about 90°" is not exact.

"Angles less than 90° vs. angles greater than 90°" is exact.

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will be "allsorted" (left-BP509).

For example:

"Angles less than 90° vs. angles greater than 90°" is not allsorted.

"Angles less than or equal to 90° vs. angles greater than 90°" is allsorted.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of these is typically only possible to notice when the previous conditions are true.

If a spectrum Bongard Problem obeys 1) and 2), then it will be "exact" (left-BP508).

For example:

"Big vs. small" is not exact.

"Angles less than 90° vs. angles greater than 90°" is exact.

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will be "allsorted" (left-BP509).

For example:

"Angles less than 90° vs. angles greater than 90°" is not allsorted.

"Angles less than or equal to 90° vs. angles greater than 90°" is allsorted.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of these is typically only possible to notice when the previous conditions are true.

If a spectrum Bongard Problem obeys 1) and 2), then it will be "exact" (left-BP508).

For example:

"Big vs. small" is not exact.

"Angles less than 90° vs. angles greater than 90°" is exact.

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will be "allsorted" (left-BP509).

For example:

"Angles less than 90° vs. angles greater than 90°" is not allsorted.

"Angles less than or equal to 90° vs. angles greater than 90°" is allsorted.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

Note there is a commonly encountered kind of "spectrum" Bongard Problem that is also "exact" (left-BP508): 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

These Bongard Problems may also be "allsorted" (left-BP509) when 3) the precise threshold value is itself sorted on one of the two sides. (This is also only possible when the previous conditions is true.)

For example, contrast:

"Angles less than or equal to 90° vs. angles greater than 90°."

"Angles less than 90° vs. angles greater than 90°."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A commonly encountered class of "spectrum" Bongard Problem that is also "exact" (left-BP508): 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A commonly encountered class of "spectrum" Bongard Problem that is also "allsorted" (left-BP509): 1) it fits the description in the previous paragraph and 2) the precise threshold value is itself sorted on one of the two sides. (The latter is only possible when the former is true.)

For example, contrast:

"Angles less than or equal to 90° vs. angles greater than 90°."

"Angles less than 90° vs. angles greater than 90°."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A common type of "exact" (left-BP508) "spectrum" Bongard Problem: 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A common type of "allsorted" (left-BP509) "spectrum" Bongard Problem: 1) it fits the above description and 2) the precise threshold value is itself sorted on one of the two sides. (The latter is only possible when the former is true.)

For example, contrast:

"Angles less than or equal to 90° vs. angles greater than 90°."

"Angles less than 90° vs. angles greater than 90°."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A continuous "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A continuous "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.)

For example, contrast:

"Angles less than or equal to 90° vs. angles greater than 90°."

"Angles less than 90° vs. angles greater than 90°."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A continuous "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A continuous "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) For example, contrast:

"Angles less than or equal to 90° vs. angles greater than 90°."

"Angles less than 90° vs. angles greater than 90°."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A continuous "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A continuous "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) For example, contrast:

"Angles less than or equal to 90 degrees vs. angles greater than 90 degrees."

"Angles less than 90 degrees vs. angles greater than 90 degrees."

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A continuous "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A continuous "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) Contrast these:

"Angles less than or equal to 90 degrees vs. angles greater than 90 degrees."

"Angles less than 90 degrees vs. angles greater than 90 degrees."

Bongard Problems about comparison of quantity vs. other Bongard Problems.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

Spectra can be either continuous or discrete.

A continuous "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A continuous "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is then compared with a fixed threshold value, separating less and greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is compared with a fixed threshold value, separating what is less and what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. (The latter is only possible when the former is true.)

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when 1) it is "exact" and 2) the precise threshold value is sorted on one of the two sides. (The latter is only possible when the former is true.) Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is compared with a fixed threshold value, separating what is less and what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. The latter is only possible when the former is true.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"); to determine whether an object fits left or right, its value is compared with a fixed threshold value, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. The latter is only possible when the former is true.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. In a "spectrum" Bongard Problem, there is an evident way to assign each object a number value (e.g. "size" or "number of holes"). To determine whether an object fits left or right, its value is compared with a fixed threshold number, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. The latter is only possible when the former is true.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are about comparison of quantity. There is an evident way to assign each object a number value (e.g. "size" or "number of holes"). To determine whether an object fits left or right, its value is compared with a fixed threshold number, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. The latter is only possible when the former is true.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are based on comparison of quantity. There is an evident way to assign each object a number value (e.g. "size" or "number of holes"). To determine whether an object fits left or right, its value is compared with a fixed threshold number, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when 1) the values assigned to objects are precise and 2) the threshold value between the two sides is precise. The latter is only possible when the former is true.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are based on comparison of quantity. There is an evident way to assign each object a number value (e.g. "size" or "number of holes"). To determine whether an object fits left or right, its value is compared with a fixed threshold number, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when the values assigned to objects are precise and the threshold value between the two sides is precise.

A "spectrum" Bongard Problem is "allsorted" (left-BP509) when it is "exact" and the precise threshold value is sorted on one of the two sides. Consider "angles less than or equal to 90 degrees vs. angles greater than 90 degrees", as opposed to "angles less than 90 degrees vs. angles greater than 90 degrees".

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are based on comparison of quantity. There is an evident way to assign each object a number value (e.g. "size" or "number of holes"). To determine whether an object fits left or right, its value is compared with a fixed threshold number, separating what is less from what is greater.

A "spectrum" Bongard Problem is "exact" (left-BP508) when the values assigned to objects are precise and the threshold value between the two sides is precise.

A "spectrum" Bongard Problem is "allsorted" (left-BP509), meaning it leaves no relevant border cases unsorted, when it is not only "exact" but also the precise threshold value is sorted on one of the two sides. Consider "angle less than or equal to 90 degrees vs. angle greater than 90 degrees", as opposed to "angle less than 90 degrees vs. angle greater than 90 degrees".

If there is a way of putting objects on a spectrum, each value x in the spectrum.

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

See BP874 for the version with pictures of Bongard Problems instead of links to pages on the OEBP.

Bongard Problems about comparison on a spectrum vs. other Bongard Problems.

Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

Many Bongard Problems are based on comparisons of quantities. It is possible to make a "greater-than-x vs. less-than-x" Bongard Problem for each value x given a spectrum of quantities. In the OEBP, we do not store all the different x values as separate BP idea pages. Instead, we have one page for the spectrum.