EQUATIONS®® Variations Always in Effect: Middle



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 and/or the Goal 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.

EQUATIONS®® Odd Year Variations: Middle


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.


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 must not be greater than 1000 or less than -1000.

EQUATIONS®® Even Year Variations: Middle



An upside down Radical means percent of. A Upside Down Radical B means A% of B when A and B are numbers. In the Goal or a Solution A and/or B may be a two-digit numeral.


Decimal Point

The Exponentiation may be used as a decimal point If so used in the Goal or a Solution, an Exponentiation may be combined in a numeral with at most three digits. When used as a decimal, Exponentiation takes precedence over all other operations. This usage of Exponentiation does not rule out Exponentiation for exponentiation. Therefore, in Solutions, players are encouraged to write a decimal point instead of Exponentiation when they want to interpret Exponentiation as a decimal point. Also one Exponentiation may be used as a decimal point and another for exponentiation.


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.



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).


Red Numeral Exponent

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

Return to the EQUATIONS®® Page.

Take some time to visit our various teams, participants, schools, and games:

If you have any questions about and/or corrections to this page or if you just want to make a comment and/or send feedback, feel free to contact Betty Smith.