EQUATIONS®® Variations Always in Effect: Junior



A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number. The reciprocal is a fraction of one over the number.



A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number. The additive inverse is the negative of that number.


Zero Wild

The 0 cube may represent any numeral on the cubes, but it must represent the same numeral everywhere it occurs (Goal and Solution). Each Solution writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in his Solution.



There are two occurrences of the factorial operator (factorial) available, like parentheses, to be used in the Solution as the Solution writer chooses to use them. All uses of factorial in the Solution must be in writing.


Base M

Both the Goal and the Solution must be interpreted as base M expressions, where the player choosing this variation specifies M for the shake as eight, nine, or ten. Two-digit numerals are allowed in Solutions.


Powers of the Base

1 (one) may represent any integral power of ten. (If 1 is used in a two-digit numeral, it stands for 1.) If base M is also chosen, 1 represents any integral power of M.


Multiple of K

A Solution must not equal the Goal but must differ from the Goal by a non-zero multiple of K, where the player choosing this variation specifies K for the shake as a whole number from six to twelve, inclusive.



The Goal and/or Solution may be or may include a three-cube expression of the form ABAddition. ABAddition is interpreted as a repeating decimal. It may be interpreted as .ABABABAB... or ABBBBBB... A player who presents a Solution is correct if the Solution satisfies either interpretation of the Goal. No decimal points may be used in the Solution (except when the decimal point variation is also chosen for the shake).

EQUATIONS®® Odd Year Variations: Junior


Multiple Operations

Every operation sign in Required or Permitted may be used many times in any Solution.


Smallest Prime

Multiplication A means "the smallest prime bigger than A," where A is a rational number less than or equal to 200. A rational number is a whole number.


Addition = Average

Addition shall not represent addition; instead it shall represent the operation of averaging two numbers.

EQUATIONS®® Even Year Variations: Junior


Number of Factors

Multiplication may be used to indicate the number of counting number factors of a counting number, including the number itself and one (that is, Multiplication A = the number of factors of A, where A is a counting number <= 200.


Red Numeral Exponent

Any red numeral may be used as an exponent without being accompanied by an Exponentiation cube.


Add to Goal

Instead of a regular move on his/her turn, a player may add a cube to the Goal. The cube may be placed anywhere in the Goal. However, the limit of five cubes in the Goal, with no numeral containing more than two consecutive digits still prevails. To use this variation, (A) A goal must be set before a challenge is made with at least one cube on the goal line; and (B) If a player states that a Goal is set before actually setting a goal (the goal line is blank), a minus 1 point is assessed.

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