ON-SETS: The Game of Set Theory®

ON-SETS® was created by by Layman E. Allen (University of Michigan), Peter Kugel (Boston University) and Martin Owens (Mitre Corporation). This intriguing game for two or three players teaches logic and set theory, which is the foundation of modern mathematics. Players learn the concepts of union, intersection, logical differences, complement, identity, inclusion, and null and universal sets.

ON-SETS® involves set manipulation and goal setting through the use of cards, cubes and a timer. Sixteen cards containing all possible combinations of blue (B), red (R), green (G), and yellow (Y) dots are dealt to form the Universe, the set of cards, which are dealt before the Goal is set. The number of cards that must be dealt depends on the player's division.

The player to the right of the Goal-setter deals the cards for the shake.

All Cards Possible in the ON-SETS® Universe

Blue Spot
Red Spot
Green Spot
Yellow Spot
Blue Spot
Red Spot
Green Spot
Blank Spot
Blue Spot
Red Spot
Blank Spot
Yellow Spot
Blue Spot
Red Spot
Blank Spot
Blank Spot
Blue Spot
Blank Spot
Green Spot
Yellow Spot
Blue Spot
Blank Spot
Green Spot
Blank Spot
Blue Spot
Blank Spot
Blank Spot
Yellow Spot
Blue Spot
Blank Spot
Blank Spot
Blank Spot
Blank Spot
Red Spot
Green Spot
Yellow Spot
Blank Spot
Red Spot
Green Spot
Blank Spot
Blank Spot
Red Spot
Blank Spot
Yellow Spot
Blank Spot
Red Spot
Blank Spot
Blank Spot
Blank Spot
Blank Spot
Green Spot
Yellow Spot
Blank Spot
Blank Spot
Green Spot
Blank Spot
Blank Spot
Blank Spot
Blank Spot
Yellow Spot
Blank Spot
Blank Spot
Blank Spot
Blank Spot

Cubes are rolled to form Resources for the Goal and Solution(s).

  • Eight of the cubes contain the same four color dots as the cards.
  • Four of the cubes have red symbols representing the four operations of Union (Union), Intersection (Intersection), Complement (Complement), and Subtraction (Subtraction).
  • Three of the cubes have blue symbols to represent the set names and relationships of Universe (Universe), Empty Set (Empty Set), Set-Inclusion (Subset), and Set-Identity (Equal).
  • Three numeral cubes contain the digits 1 to 5 with which players formulate a goal.

Students play in groups of two or three. Each player rolls a digit cube to see who has the highest number. That player gives the first variation, rolls the cubes, and sets the goal. The player to the right lays out the universe. The player to the left will be the second player.

To make the game more interesting and challenging, special rules called Variations are in force in each division: Elementary, Middle, Junior, and Senior. Before the cubes are rolled, each player selects a variation from the list for the division. Beginning with the first player, the players name one variation apiece for a total of three. In a two-player game, the second player will name two variations.

Within the first two minutes, the first player uses the digit cubes to set a Goal, which is the number of cards that must be in the solution set. Each player must use the symbols rolled to form a set equal to the Goal. For example, a Solution for a Goal of 5 might be B Union R ("blue union red"), which represents a set of five cards, which contain either a blue OR a red dot.

Goals in ON-SETS®

The player who rolls the cubes must set a Goal by transferring the cube(s) of the Goal from Resources to the Goal section of the playing mat. A Goal consists of at least one and at most three digit cubes that form an expression that names a whole number.

If more than one cube is used to set the Goal, the way the cubes are placed in the Goal determines the Goal's value.
  • The sum of two numbers is indicated by placing the cubes in a horizontal line (side by side).
  • The product of two numbers is indicated by placing the cubes in a vertical line.
  • The negative of a number is indicated by placing the cube so that its numeral is upside-down.
The following are the only legal configurations for the Goal.


GOAL


MEANING

 

GOAL


MEANING

AB
A + B
A
B
C
A x B x C
ABC
A + B + C
A
BC
A (B+C) or
AB + AC
A
B
A x B
A   
BC
(AB) + C

The second player has two minutes to make his/her first move. During subsequent plays, students take one-mintue turns moving cubes to the Required, Permitted, or Forbidden sections of the playing mat pictured below until a Challenge can be made.

The ON-SETS® Playing Mat

RESOURCES

FORBIDDEN

PERMITTED

REQUIRED


Goal _______________

The playing mat for ON-SETS® consists of five sections.

  1. Resources: the cubes are placed here after they have been rolled by the Goal-setter.
  2. Goal: the Goal-setter places the Goal on this section.
  3. Required: cubes played here must be used in any Solution.
  4. Permitted: cubes played here may be used in any Solution.
  5. Forbidden: cubes played here may not be used in any Solution.
Any cube moved to Required must be used in any Solution; any cube in Permitted may be used; any cube in Forbidden may not be used. Thus, the players themselves shape the Solution, forcing one another to create new Solutions in response to moves. ON-SETS® forces players to revise their solutions in response to opponents' moves. Quick and logical thinking is required to win the match.

To challenge, a player must pick up the challenge block and state the challenge.

  • Challenge Win means there are enough cubes available on the mat to solve by using only one more cube. The challenger must solve. The mover may not solve. The third player must Side with either the mover or challenger within the first minute. The challenger has the burden of proof to provide an Avoid Move that the last player could have chosen that would not have made the solution possible or impossible.
  • Challenge Impossible means that no solution is possible no matter how many cubes are used from resources. The last player to move must solve. The challenger may not solve, and the third player must Side with either the mover or challenger within the first minute. All cubes from resources and the Permitted and Required sections of the playing mat may be used to write a solution.
  • Challenge Trap means that the challenger feels a player before him/her should have called Challenge Win or Challenge Impossible but did not. The cubes that have been played, since that challenge should have been made, are moved back into resources, and the players solve as if Challenge Win or Challenge Impossible has been called.
  • Forceout means that there is nothing the mover can do to keep someone from calling a Challenge Win after his/her move. The player moves a cube to the mat, and the next player calls Forceout instead of Challenge Win because there was no way to Avoid making it possible to solve with one more cube. All players who agree write a solution. Those who disagree may Challenge the Forceout.
ON-SETS® is scored like this:
  • The player who wins the Challenge scores 10 points.
  • The loser of the Challenge scores 6.
  • If there is a third player, he/she must side with or against the Challenger and scores points depending upon that decision. If the player correctly sides with the Mover, he/she earns 10 points because the Challenger was incorrect. If the player correctly sides with the Challenger, he/she only earns 8 points because that player could have been the first correct Challenger.

Solutions in ON-SETS®

To be correct, a Solution must be a legal expression that also satisfies the following criteria.
  1. The Solution contains a valid Set-Name part.
  2. The Solution equals the Goal. That is, the number of cards selected from the Universe by the Set-Name equals the Goal.
  3. The Solution uses the cubes correctly.
  4. After a Challenge Win, the Solution must contain at most one cube from Resources.
  5. After a Challenge Impossible, any cubes in Resources are considered to be in Permitted.
  6. The Solution satisfies all conditions imposed by the variations selected for that shake.
  7. 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 (Union, Intersection, , 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.
  8. Rules for Restriction Statements for Middle, Junior, Senior Divisions only:
    • The Solution must contain a Restriction part if there are one or more Equal or Subset cubes in Required.
    • If no Equal or Subset cubes are in Required but some are in Permitted or Resources, the Solution may include a Restriction part.
    • 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.
  9. The Solution uses no cube in Forbidden.
  10. Every cube in Required must be used in the Restriction part (if there is one). These same cubes (except any Equal or Subset) must also be used in the Set-Name.
  11. Each cube in Permitted may be used in the Restriction part (if there is one). These same cubes (except any Equal or Subset) may also be used in the Set-Name.

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