EQUATIONS: The Game of Creative Mathematics®
EQUATIONS®®, a math game for two or three players who use cubes to make goals and find solutions to those goals, was created in 1965 by Layman E. Allen, Professor of Law and Senior Research Scientist, University of Michigan. The basic game of EQUATIONS®® uses the arithmetic operations of addition, subtraction, multiplication, division, exponents, and roots. In Adventurous EQUATIONS®®, advanced variations cover further operations, various number bases and higher mathematics. The game can be as simple or complex as the players make it, depending on their mathematical knowledge. This creative prelude to algebra assists players in developing an understanding of number concepts, number systems, factors and primes, order of operations, and simple algebraic equations.
Special rules called Variations, which change every other year, are in force in each division: Elementary, Middle, Junior, and Senior. The variations reinforce what is taught in the mathematics curriculum. Before the cubes are rolled, each player selects a variation from the list provided for that division. EQUATIONS® encourages students to apply the mathematics they already know and learn concepts they will not meet in the curriculum for several years.
The EQUATIONS®® game consists of a playing mat and 24 cubes, which each contain four digits and two operation signs (+, -, X, ÷, *, or ).
The EQUATIONS®® Playing Mat
The playing mat for EQUATIONS®® consists of five sections.
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. EQUATIONS®® forces players to revise their solutions in response to opponents' moves. Quick and logical thinking is required to win the match.
the cubes are placed here after they have been rolled by the Goal-setter.
- Goal: the Goal-setter places the Goal on this section.
- Required: cubes played here must be used in any Solution.
- Permitted: cubes played here may be used in any Solution.
- Forbidden: cubes played here may not be used in any Solution.
A numerical Goal is set and players must form a Solution equal to the Goal from the cubes rolled (the Resources). For example, the Goal might be 2 * 5 (2 to the fifth power, which is 32). A Solution might be: (5 X 5) + 8 - 1
The EQUATIONS®® Goal
The player who rolled the cubes begins play by setting the Goal, which must obey the following rules.
To challenge, a player must pick up the challenge block and state the challenge.
- It must represent a valid numerical expression.
- It must use only one- or two-digit numbers.
- It must consist of from one to five cubes, inclusive.
- The * cube is used for raising to a power--not multiplication.
- The radical sign is used for taking a root. If no number appears in
front of the as an index, it is understood to mean square root.
- The + sign means addition. It is not a positive sign.
- The - sign means subtraction. It is not a negative sign.
- The Goal may be ambiguous (have more than one interpretation) like 3+4x2.
In this case the Goal-setter may group the cubes in the Goal to force an
interpretation. If not, a Solution may equal any legal value of the
- Suppose the Goal is (3 + 4) x 2 [with the parentheses indicating
that the Goal-setter groups the 3 + 4 on the mat apart from the x 2].
- Then any Solution must equal 14.
- If the Goal is 3 + (4 x 2), then the only value is 11.
- However, if the Goal is 3+4x2 [with no grouping], a Solution may equal either 14 or 11.
- 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.
Solutions in EQUATIONS®®
To be correct, a Solution in EQUATIONS®® must be a legal expression that also satisfies the following criteria.
EQUATIONS®® is scored like this:
- The Solution equals a legal interpretation of the Goal.
- The Solution uses the cubes correctly.
- The Solution uses all the cubes in Required.
- The Solution uses no cube in Forbidden.
- The Solution may use one or more cubes in Permitted.
- After a Challenge Win, the Solution must contain at most one cube from Resources.
- After a Challenge Impossible, any cubes in Resources are considered to be in Permitted.
- The Solution contains only one-digit numerals unless two-digit Variation is called in Elementary Division or a base M Variation is called in Middle, Junior, or Senior Divisions..
- The Solution satisfies all conditions imposed by the variations selected for that shake.
- Every legal interpretation of the Solution equals a value of the Goal.
- In Adventurous EQUATIONS®® the order of operations of mathematics does not apply to Solutions.
- 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 where they will create a wrong interpretation.
- If this revised Solution does not equal any value of the Goal, the Solution is incorrect.
Please visit Dr. Layman E. Allen's EQUATIONS®: the Game of Creative Mathematics Page.
- 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.