### On-Sets® Solutions

#### To be correct, a Solution must be a legal expression that also satisfies the following criteria.
- The Solution contains a valid Set-Name part.
- Middle, Junior, Senior Divisions only: The Solution contains a Restriction part if there are one or more = or C cubes in Required.

##### If no = or C cubes are in Required but some are in Permitted or Resources, the Solution *may* include a Restriction part.

- The Solution equals the Goal. That is, the number of cards selected from the Universe by the Set-Name equals the Goal.
##### If the Solution includes one or more Restrictions, these must be applied to the Universe *before* the Set-Name is worked out. They may be applied to the Universe in any order.

- The Solution uses the cubes correctly.
- The Solution contains at least two cubes.
- Every cube in Required is used in the Restriction part (if there is one). These same cubes (except any = or C) must also be used in the Set-Name.
- Each cube in Permitted may be used in the Restriction part (if there is one). These same cubes (except any = or C) may also be used in the Set-Name.
- The Solution uses
*no* cube in Forbidden.

- After a Now challenge, the Solution must contain
*at most one* cube from Resources.

- After a Never challenge, any cubes in Resources are considered to be in Permitted.

- The Solution satisfies all conditions imposed by the variations selected for that shake.
- Every legal interpretation of the Solution equals the Goal.

##### An ambiguous Solution is one that has more than one legal interpretation. Such a Solution is incorrect if an opponent shows that one of the interpretations does not equal the Goal.

The only defined order of operations in On-Sets is that the ' operation takes priority over all other operations (U, ñ, -, and special operations defined by variations). Consequently a Solution may be ambiguous if the writer does not use parentheses (or other symbols of grouping such as brackets or braces) to indicate the order of operations.

If an opponent believes there is an interpretation of a Solution which does not equal the Goal, that opponent should copy the Solution on his paper and add symbols of grouping to create a wrong interpretation. If this revised Solution does not equal the Goal, the Solution is incorrect.