# Equations Variations Elementary Equations® Variations

Note {counting numbers} = {natural numbers} = {positive integers} = {1, 2, 3, 4, …}            {whole numbers} = { 0, 1, 2, 3, 4, …}

GENERAL RULE     If * is used for raising to a power, both base and exponent must be whole numbers. If √ is used for the root operation, the index must be a counting number, and the base and total value must be whole numbers.

1. Sideways A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number.
2. Upside-down A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number.
3. 0 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 Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation.
4. Factorial There are two occurrences of the factorial operator (!) available, like parentheses, to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing.
5. Two-digit numerals Two-digit numerals are allowed in Solutions.
6. Three-operation Solution Any Solution must contain at least three operation symbols. The operation symbols are +, –, x, ÷, *, and √ .
7. Smallest prime xA means “the smallest prime bigger than A,” where A is a rational number less than or equal to 200.
8. Percent (upside-down radical) means “percent of.” That is, A  B = A% of B where A and B are numbers. In the Goal or Solution, A and/or B may be a two-digit numeral.
9. Decimal point * may represent a decimal point. If so used in the Goal or Solution, an * may be combined with at most three digits to form a numeral. When used as a decimal, * takes precedence over all other operations.

Middle Equations® Variations

1. Sideways A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number.
2. Upside-down A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number.
3. 0 wild The 0 cube may represent any symbol on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Equation.
4. Factorial There are two occurrences of the factorial operator (!) available, like parentheses, to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing.
5. 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.
6. 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 eleven, inclusive. The Goal must not be greater than 1000 or less than –1000.
7. Percent (upside-down radical) means “percent of.” That is, A  B = A% of B where A and B are numbers. In the Goal or Solution, A and/or B may be a two-digit numeral.
8. Decimal point * may represent a decimal point. If so used in the Goal or Solution, an * may be combined with at most three digits to form a numeral. When used as a decimal, * takes precedence over all other operations.
9. AB+ The Goal or Solution may be or may include a three-cube expression of the form AB+ which is interpreted as a repeating decimal. AB+ must represent either .ABABAB… or .ABBBBB
10. Exponent Any numeral on a  cube may be used as an exponent without being accompanied by an * cube. The player selecting this variation fills the blank in the previous sentence with one of the colors red, blue, green, or black.

Junior Equations® Variations 2007-8

SPECIAL RULE: The following two variations are in effect for all shakes.

1. Sideways A cube representing a non-zero number may be used sideways in the Goal or Solution to equal the reciprocal of that number.
2. Upside-down A cube representing a number may be used upside-down in the Goal or Solution to equal the additive inverse of that number.

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1. 0 or x wild The 0 or x cube may represent any symbol on the cubes, but it must represent the same symbol everywhere it occurs (Goal and Solution). Each Equation-writer must specify in writing the interpretation of the 0 or x cube if it stands for anything other than itself in the Equation. The player selecting this variation specifies whether 0 or x (but not both) is wild for the shake.
2. 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.
3. 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, eleven, or twelve. Two-digit numerals are allowed in Solutions. For bases eleven and twelve, * may be used for the digit ten; in base twelve √ may be used for the digit eleven.
4. 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 eleven, inclusive.
5. Multiple operations Every operation sign in Required or Permitted may be used many times in any Solution. If the factorial variation is also chosen for the shake, an unlimited number of factorial operators may be used in each Solution. At most two factorials may be used in the Goal.
6. Factorial There are two occurrences of the factorial operator (!) available, like parentheses, to be used in the Solution and/or the Goal as the Equation-writer chooses to use them. All uses of ! in the Equation must be in writing. However, if multiple of k is also chosen for the shake, no factorial may be placed in the Goal.
7. Number of factors xA means “the number of counting number factors of A,” where A is a counting number.
8. Exponent Any numeral on a  cube may be used as an exponent without being accompanied by an * cube. The player selecting this variation fills the blank in the previous sentence with one of the colors red, blue, green, or black.

Senior Equations® Variations

All the Junior variations for this year plus the following:

11. √=i shall not represent the root operation but instead may represent the imaginary number i (such that i2 = –1). The √ may be placed immediately before or after a numeral without the x sign.

1. Decimal in Goal Each Equation-writer may determine where decimal points occur in the Goal.

1. Log A sideways ÷ represents the log operation. Thus if a and b are positive real numbers (b ≠ 1), a b equals logba.

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