# Basic Equations Rules Basic Equations can be introduced with the following seven rules. The complete set of the Official Tournament Rules should be used once players become familiar with the game.

# I.      GOAL Rule

Two- or three-player matches will be played. To start, one player rolls 24 cubes (six of each color). The numerals and operations that show on the top faces of the cubes form the Resources.

1. The player who rolled the cubes sets a Goal by moving one to six cubes from the Resources to form a numerical expression on the Goal section of the playing mat.
2. The Goal-setter indicates the Goal is complete by saying “Goal.”

# II.     MOVE Rule

After the Goal has been set, play progresses to the left. When it is your turn to play, you must either challenge (see rule IV) or move a cube from Resources to the Forbidden, Permitted, or Required section.

# III.    SOLUTION Rule

A Solution, which is written on paper, must equal the Goal and also obey these requirements.

1. It must use the cubes as specified on the playing mat: all cubes in Required, no cubes in Forbidden, and none, some, or all of the cubes in Permitted.
2. It may contain only one-digit numerals. (Two-digit numerals are permitted in the Goal.)
3. It may contain parentheses to show the order of operations. (If you are not playing variations, the normal order of operations of mathematics applies.)
4. It may use the + and – signs to indicate only the operations of addition and subtraction. The x, ÷, and * signs indicate multiplication, division, and exponentiation, respectively. (This also applies to the Goal.)
5. If the radical sign (√) is used, it must be preceded by an expression to denote its index with the exception that when no index is shown, it is understood to be 2. (This also applies to the Goal.) The Solution must contain at least two cubes.

# IV.    CHALLENGE Rule

1. Whether or not it is your turn, you may challenge another player who has just set the Goal or moved. To do so, you must pick up the challenge block and say one of the following.
1. NOW (Now possible): This means that the Challenger claims a Solution can be written by using: all the cubes in Required and
1. none, some, or all of the cubes in Permitted and
2. one more cube from Resources, if needed.
2. NEVER (Never possible): This means the Challenger claims that nobody can write a Solution that satisfies all the restrictions placed on the cubes in the Forbidden and Required sections, no matter how many cubes might still be used from the Permitted section and the Resources.
2. After a challenge in a three-player match, the Third Party (the player who is neither the Challenger nor the last Mover) must agree or disagree with the Challenger.
1. If the Third Party agrees with the Challenger after a NOW challenge, the Third Party must also independently write a Solution.
2. If the Third Party disagrees with the Challenger after a NEVER challenge, the Third Party must also independently write a Solution.

# V.     CORRECTNESS Rule

After a challenge, a player is correct if and only if that player

1. has to write a Solution and does so correctly or
2. does not have to write a Solution (someone else does) and nobody writes a correct Solution.

# VI.    CHALLENGE-SCORING Rule

1. Whoever is incorrect scores 2.
2. Whoever is correct scores 6, unless he agrees with the Challenger, in which case he scores 4.

# VII.   LAST CUBE Rule

As soon as there is only one cube left in Resources, the only challenge allowed is NEVER. If nobody challenges NEVER, the player whose turn it is must move that cube to the playing mat. Then, unless someone makes a NEVER challenge, all players write a Solution. Whoever is correct scores 8 points; whoever is incorrect, scores 6.

EQUATIONS® Tournament Rules

1. Starting a Match (Round)
1. Two- or three-player matches will be played. A match is composed of one or more shakes. A shake consists of a roll of the cubes and the play of the game. A shake ends with at least one player attempting to write a Solution, which is a mathematical expression that equals the Goal and correctly uses the cubes on the playing mat. The following equipment is needed to play the game.
1. 24 cubes: there are six of each color (red, blue, green, and black). Every face of each cube contains either a digit (0, 1, 2, …, 9) or an operation sign (+, –, x, ÷, *, √).
2. A playing mat: this contains four sections.
1. Goal: cubes played here form the Goal.
2. Required: all cubes played here must be used in any Solution.
3. Permitted: any or all cubes played here may be used in any Solution.
4. Forbidden: no cube played here may be used in any Solution.

Comment Many game boards have a section labeled “Resources.” However, any reference in these rules to the “playing mat” or “mat” does not include the Resources section.

1. A one-minute sand timer: this is used to enforce time limits.
1. Players may use only pencils or pens, blank paper, and (for Adventurous Equations) variation sheets. No prepared notes, books, tables, calculators, cell phones or other electronic devices may be used. In Elementary and Middle Divisions, players may use the chart shown in Appendix B for recording the Resources, variations, Goal, and Solutions.
2. The Goal-setter for the first shake is determined by lot. On each subsequent shake, the Goal-setter is the player immediately to the left of the previous Goal-setter.

To determine the first Goal-setter, each player rolls a red cube. The player rolling the highest digit sets the first Goal. A player who rolls an operation sign is eliminated unless all players roll an operation sign. Players tied for high digit roll again until the tie is broken.

1. Starting a Shake
1. To begin a shake, the Goal-setter rolls all 24 cubes. The symbols on the top faces of the rolled cubes form the Resources for the shake.
1. A shake begins as soon as the timing for rolling the cubes is started or the cubes are rolled.
2. During a shake, no player may turn over a cube or obstruct the other players’ view of any cube. (See section IX-C.)
2. In Adventurous Equations, after the cubes are rolled but before the Goal is set, each player must select a variation from the appropriate list in section XIII of these rules. A variation is a special rule which, if it conflicts with any of the regular tournament rules, supersedes those rules.
1. The Goal-setter makes the first selection, then the player to the left of the Goalsetter, then the third player if there is one.
1. Each player has 15 seconds to make a variation selection.
2. To begin a shake, the Goal-setter has one minute to roll the cubes. At the end of this minute, he has 15 seconds to select a variation. However, if the Goal-setter selects a variation before the minute for rolling the cubes expires, the next player has the rest of that minute plus 15 seconds to select a variation. If the second player also selects a variation before that minute expires, the third player (if there is one) has the rest of that minute plus 15 seconds to select.
3. A player selects a variation by circling its name in the list for that shake. This list is located on the reverse side of the scoresheet or on a separate sheet. For certain variations (e.g., Base or Multiple of k), the player must also fill in a blank to indicate which base or value of k is chosen, and so on.
2. If a player selects a variation that has no effect on the shake, a variation that conflicts with one already chosen for the shake, or a variation that has already been chosen for the shake, the player loses one point and must pick another variation. If, on the second try, the player still does not select an appropriate variation, he loses another point and may not pick a variation for that shake.

If a player’s illegal variation selection is not pointed out before the next player selects a legal variation or a legal Goal is set (whichever comes first), the player making the illegal selection is not penalized. However, the illegal variation is ignored for the shake.

Examples It is illegal to choose 0 wild when no 0 cube is in Resources or Average when no + was rolled.

1. In two-player matches in Elementary, Middle, and Junior Divisions, the player who is not the Goal-setter must select two variations for the shake. In Senior Division, any player may pick two variations for any shake in either a two- or threeplayer match.
• Legal Mathematical Expressions
1. A legal mathematical expression is one which names a real number and does not contain any symbol or group of symbols which is undefined in Equations.

Example     a ÷ 0 for any value of a does not name a real number. (See section C below for additional examples.)

Comment An expression written on paper may contain pairs of grouping symbols such as parentheses, brackets, or braces even though these do not appear on the cubes. These symbols indicate how the Equation-writer would physically group the cubes if the Solution were actually built with the cubes.

1. The symbols on the cubes have their usual mathematical meanings with the following exceptions.
1. The + and – cubes may be used only for the operations of addition and subtraction; they may not be used as positive or negative signs.

Examples +7, –8, 6x+4, and 17÷ (–8) are illegal expressions.

1. If the radical sign (√) is used, “it must always be preceded by an expression to denote its index” unless the index equals 2. If no index is shown, it is understood to be 2.

Examples 2√9 or just √9 is legal and means “the square root of 9”; 1√2 means “the first root of 2,” which is 2. (2+1)√8 means “the cube root of 8,” which is 2. 4x√9 means “4 times the square root of 9,” which is 4 x 3 or 12. 3√√9 means “the cube root of the square root of 9,” which is the sixth root of 9. (This expression is illegal in Elementary Division – see the “General Rule” in section XIII-A.)

1. * means exponentiation (raising to a power).

Example     4*2 means 42, which is 4 x 4 =16.

1. Expressions involving powers and roots must satisfy these requirements.
1. Even-indexed radical expressions indicate only non-negative

Examples 2√9 or just √9 equals 3, not –3; 4√16 = 2 (not –2).

1. The following expressions are undefined.
1. 0√a where a is any number
2. 0 * a where a ≤ 0
3. a b where a is an even integer and b is negative
4. (a ÷ b) √ c where c is negative and, when a ÷ b is reduced to lowest terms, a is an even integer and b is an odd integer
5. a * (b ÷ c) where a is negative and, when b ÷ c is reduced to lowest terms, b is an odd integer and c is an even integer

Examples (Middle, Junior, and Senior Divisions only)

• (–8)4/6 is defined, as shown by the following steps. First reduce the fractional exponent to lowest terms: (–8)4/6 = (–8)2/3. (–8)2/3 is of the form a * (b ÷ c) where a is negative. Since b is even and c is odd, (–8)2/3 is defined. (–8)2/3 = 3√(–8)2 = 3√64 = 4.
• (–4)2/4 is not defined because (–4)2/4 = (–4)1/2, which is of the form a * (b ÷ c) with a negative, b odd, and c

Note The following reasoning is not allowed since the exponent is not reduced first:       (–4)2/4 = 4√(–4)2 = 4√16 = 2.

• 3/6√(–9) is defined because 3/6√(–9) = 1/2√(–9), which is of the form (a ÷ b) √ c, with c However, a is odd and b is even. So 3/6√(–9) = (–9)2 = 81.
• 8/2√(–5) is not defined because 8/2√(–5) = 4√(–5), which is of the form ab where a is even and b is negative.
1. Setting the Goal
1. 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.
2. A Goal consists of at least one and at most six cubes which form a legal expression.
1. Numerals used in the Goal are restricted to one or two digits. The use of operation signs is optional.

Examples of legal Goals 6, 8–9, 17×8, 9+8–5, 87÷13, 3√64, √49

Examples of illegal Goals 125 (three-digit numerals not allowed), 23+18+7 (too many cubes), 45x (does not name a number), +8 (does not name a number).

1. The order of operations of mathematics does not apply to the Goal. The Goalsetter may physically group the cubes in the Goal to indicate how it is to be interpreted. If the Goal-setter does not group the cubes, the Goal may be interpreted in any valid way.

Examples

1. 2x 3+5 (with space between x and 3) means 2 x (3 + 5).
2. 2×3 +5 (with space between 3 and +) means (2 x 3) + 5.
3. The Goal 2×3+5 (with no spaces) may be interpreted as either 2 x (3 + 5) or (2 x 3) + 5. Comment The Goal-setter may not be able to remove all ambiguities from the Goal. Example where the Goal-setter wants to apply the √ to the entire expression (5+4)x9. Declaring orally that the √ applies to everything that follows is not binding.

Players may interpret this Goal as [√(5+4)]x9 or as √[(5+4)x9]. If the Goal is √5+4  x9 (no space after the √), it is not ambiguous and must be interpreted as [√(5+4)]x9. The same would apply if the √ were an x for the Number of factors variation (or Smallest prime in Elementary – see section XIII below).

1. Once a cube touches the Goal section of the mat, it must be used in the Goal.
1. The Goal-setter indicates the Goal has been set by saying “Goal.”
2. The Goal-setter may rearrange or regroup the cubes in the Goal section until he says “Goal.”
3. The Goal may not be changed once it has been set.
2. Before moving the first cube to the Goal section of the mat, the Goal-setter may make a bonus move.
1. To make a bonus move, the Goal-setter must say “Bonus,” then move one cube from Resources to Forbidden.
2. A Goal-setter who is leading in the match may not make a bonus move.

If the Goal-setter makes a bonus move while leading in the match and an opponent points out the error before the next player moves or someone legally challenges, the cube in Forbidden is returned to Resources. In Junior and Senior Divisions, the Goal-setter is also penalized one point. D. If the Goal-setter believes no Goal can be set which has at least one correct Solution (see section VII), he may declare “no Goal.” Opponents have one minute to agree or disagree with this declaration.

1. If all players agree, that shake is void and the same player repeats as Goal-setter for a new shake. Comments
1. The Goal-setter would declare “no Goal” only in those rare instances when an unusual set of Resources was rolled. For example, there are less than three digit cubes or only one or two operation cubes. (Even in these cases, the Goal-setter could choose a variation like 0 wild that might allow a Goal to be set.)
2. Players receive no points for the void shake.
3. If an opponent does not agree with the “no Goal” declaration, he indicates disagreement by picking up the challenge block (see section VI-B). He then has one minute to set a Goal. If he does, the original Goal-setter for the shake receives a two-point penalty unless a correct Never challenge (see section VI-A) is made against this Goal before the next player moves a cube to the mat. However, if the disagreeing player decides to say “no Goal,” then he loses a point, the shake is void, and the original Goal-setter rerolls the cubes for a new shake.
2. Moving Cubes
1. “After the Goal has been set, play progresses in a clockwise direction” (to the left).
2. When it is your turn to play, you must either move a cube from Resources to one of the three sections of the playing mat (Required, Permitted, Forbidden) or challenge the last Mover.

The move of a cube is completed when it touches the mat. Once a cube is legally moved to the mat, it may not be moved again during the shake.

1. If you are not leading in the match, then “on your turn you may take a bonus move before making a regular move.”
1. To make a bonus move, the Mover must say “Bonus,” then move one cube from Resources to Forbidden.

1. “If you do not say ‘Bonus’ before moving the cube to Forbidden, the move does not count as a bonus move but as a regular move to Forbidden.” You are not entitled to play a second cube.
2. When making a bonus move, the first cube must go to Forbidden. The second cube may be moved to Required, Permitted, or Forbidden.
1. If the player in the lead makes a bonus move and an opponent points out the error before another player makes a legal move or challenge, the Mover must return the second cube played on that turn to Resources. In Junior and Senior Divisions, the Mover also loses one point.
1. Challenging
1. “Whether or not it is your turn, you may challenge another player who has just completed a move” or set the Goal. The only two legal challenges are Now and Never.
1. By challenging Never, a player claims that no correct Solution can be written regardless of how the cubes remaining in Resources may be played.

1. If the Goal is not a legal expression, an opponent should challenge Never. Examples of such Goals are +8, 65+87–3, 122, and so on.
2. Occasionally it is obvious before the Goal-setter completes the Goal that no Solution is possible. Examples are using more than six cubes in the Goal or (in Mid/Jr/Sr) is using an 8 or 9 in the Goal when Base eight was called. However, opponents must still wait until the Goal-setter indicates the Goal is finished before challenging. You may not pick up the challenge block and “reserve” the right to challenge when the Goal is completed.
1. By challenging Now, a player claims that a Solution can be written using the cubes on the mat and, if needed, one cube from Resources.
1. A player may challenge Now only if there are at least two cubes in Resources.

If a player challenges Now with fewer than two cubes in Resources, the challenge is invalid and is set aside. The challenger is also penalized one point. (See section B below.)

Comment If only one cube remains in Resources and no one challenges Never, then a Solution is possible using that one cube. Since the latest Mover had no choice but to play the second-to-last Resource cube to the mat, it is not fair that he be subject to a Now challenge. (However, a Never challenge could be made.) See section VIII for the procedure to be followed when one cube remains in Resources.

1. Since a correct Solution must contain at least two cubes, it is illegal to challenge Now after the Goal has been set but before a cube has been played to Required or Permitted.

If a player does so, that player is penalized one point and play continues.

1. A challenge block is placed equidistant from all players. To challenge, a player must pick up the block and say “Now” or “Never.”

A player who picks up the block and does not make a valid challenge is penalized one point and the challenge is set aside. Examples of invalid challenges are (a) a player attempting to challenge himself, (b) a player challenging Now when less than two cubes remain in Resources, and (c) a player picking up the block, then deciding not to challenge (without saying “Now” or “Never”). In case (c), apply the time limit rule (XI). The player has 15 seconds to state a valid challenge. If he does not do so, he loses one point and has a minute to state a valid challenge. At the end of that minute (or when the player indicates he does not want to challenge), he loses another point and play continues. Comments

• The main purpose of the block is to determine who is the Challenger in a three-player match when two players wish to challenge at the same time. The challenge block should be a cube or similar object and not a flat object like a coin. It should not be so large that two players can grab it simultaneously.
• Touching the challenge block has no significance. However, players may not keep a hand, finger, or pencil on, over, or near the block for an extended period of time. (See section IX-C.)
• A player must not pick up the challenge block for any reason except to challenge. For example, don’t pick it up to say “Goal” or to charge illegal procedure.
• Writing and Checking Solutions
1. After a valid challenge, at least one player must write a Solution.
1. After a Now challenge, the Challenger must write a Solution. (The Mover may not present a Solution.)
2. After a Never challenge, the Mover must write a Solution. (The Challenger may not present a Solution.)
3. After any challenge in a three-player match, the Third Party must decide whether to agree with the Challenger or the Mover. If the player with whom the Third Party agrees must write a Solution, then the Third Party must also write a Solution.
2. To be correct, a Solution must be a legal expression (see section III) that satisfies the following criteria.
1. The Solution must be part of a complete equation in this form.

Solution  =  Goal

Comment While Solution = Goal is the recommended form for writing the Solution, Goal = Solution is acceptable. (See Appendix A for all matters involving how Solutions are written.)

1. The Solution must equal the interpretation of the Goal that the Equation-writer presents with the Solution.

Examples

 Goal Sample Equation Goal Sample Equation 37 (6×6) + 1 = 37 11+5 (3×2)+(5×2) = 11+5 3×5+2 (5*2) – 4 + 0 = 3x(5+2) 3x  5+2 (5×4)+1 = 3x(5+2) 03 (5×5) – 2 = 03 (with 0 wild)↑                0 0 0+3 1*(7+8-4×3) = 0+3 (0 wild, Upside-down)↑ud 2 30×7 [(5*2)x8] + 6 + 4 = 30×7 (with 0 wild, 0 defaults to 0) 9÷8 5 + 4 = 9! ÷ 8!with Factorial variation

Note: See Appendix A for a more complete list of ways of indicating what ambiguous cubes (such as wild cubes) represent in Solutions and Goals. The Appendix also lists the default values of ambiguous symbols if an Equation-writer does not indicate the interpretation. However, there is no default order of operations in Equations (except when the Factorial and Exponent variations are played – see section 6b below).

1. An Equation-writer who does not write which interpretation of the Goal the Solution equals, even when there is only one interpretation, is automatically incorrect.
2. The Equation-writer does not write the value of the Goal except in those cases where writing the Goal is the same as writing its value. Examples
• The Goal is 37, or the Goal is 40 with 0 wild and the writer writes 45 to indicate what 0 represents
• For a Goal like 3×5+2, the writer must write either (3×5)+2 or 3x(5+2) and not 17 or 30.
3. If the Goal is grouped, as in the example 3x 5+2 on the previous page, an Equation-writer must write 3x(5+2) and not 3x  5+2 (with space between x and 5). 3x  5+2 (with space after x but no parentheses) does not qualify as legal grouping in a Solution and therefore does not correctly group the Goal.
1. The Solution uses the cubes correctly.
1. The Solution contains at least two
2. The Solution uses all the cubes in Required.
3. The Solution uses no cube in Forbidden.

Comment  “Since several Resource cubes may show the same symbol, it is possible to have a 2 in Forbidden which must not be used in the Solution at the same time that there is a 2 in Required which must be used.”

1. The Solution may use one or more cubes in Permitted.
2. After a Now challenge, the Solution must contain at most one cube from Resources.
3. After a Never challenge, any cubes in Resources are considered to be in Permitted.
1. The Solution contains only one-digit numerals.

Comment Certain variations (see section XIII) allow exceptions to this rule; for example, Twodigit numerals in Elementary Division and Base m in Middle/Junior/Senior.

1. In Adventurous Equations, the Solution satisfies all conditions imposed by the variations selected for that shake. (See section XIII for a list of the variations.)

Examples

• If the Elementary variation Three-operation Solution has been chosen, any Solution that contains fewer than three operations is incorrect.
• In Middle, Junior, and Senior, the Multiple of k variation requires that any Solution not equal the Goal. Instead the Solution must differ from the Goal by a multiple of k.
1. The Solution is not ambiguous. An ambiguous Solution is one that has more than one legal interpretation. Such a Solution is incorrect if an opponent shows that one of its values does not equal the interpretation of the Goal provided with the Solution.

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.

• For the procedure to be followed when a Solution-checker thinks a Solution is ambiguous, see below.
• For the Factorial and Exponent variations (see section XIII below), the mathematical order of operations will apply as follows.
• •• Factorial: In the absence of grouping symbols, ! applies to just the numeral in front of it.

Examples 4 + 7! means 4 + (7!). An opponent may not interpret it as (4 + 7)!

3√5! means 3√(5!), not (3√5)!

Suppose the Goal is 4 + 7. If an Equation-writer wants 4+(7!), just write 4+7! If the writer wants 11!, write (4+7)!

With Number of factors, 4+x7! means 4+x(7!), not 4+(x7)!

• •• Exponent (Middle/Junior/Senior only): The exponent of the selected color applies to just the numeral in front of it unless grouping symbols are used.

Examples 4+32 means 4 + (32). An opponent may not interpret it as (4+3)2.

√5! 2 means √(5!2), not (√5!)2.

With Number of factors, 4+x122 means 4+x(122), not 4+(x12)2.

With Factorial, 4+3!2 means 4+(3!)2.

1. After the time for writing Solutions has expired (or when all Equation-writers are ready), each Solution that is presented must be checked for correctness.
2. After a challenge in a three-player match (and before any Solution is presented), the Third Party must indicate by the end of the two minutes for writing Solutions whether he is presenting a Solution.

Comment To indicate his intention on the challenge, the Third Party may:

• state whether or not he will present a Solution;
• indicate which party, Mover or Challenger, the Third Party is “joining” (agreeing with) on the challenge. This can be done verbally or by pointing to the party.
• present or not present a Solution when the time limit for writing Solutions expires. In any case, the Third Party may not retract his decision once he has indicated whether or not he will present a Solution.
1. All Solutions must be presented before any is checked.
1. Once a player presents a Solution to the opponent(s), he may make no further corrections or additions even if the time for writing Solutions has not expired.
2. Each Equation-writer must circle the Solution to be checked, including the interpretation of the Goal. A writer who forgets to circle the Solution must do so immediately when asked by an opponent.
2. Opponents have two minutes to check each Solution. When more than one Solution must be checked, they may be checked in any order. In a three-player match, both opponents must check a player’s Solution during the same two minutes. No other Solution should be checked during this time.

Comment When both players in a two-way match present Solutions after the last cube has been moved (see section VIII below), only one Solution should be checked at a time.

1. Within the time for checking a Solution, opponents must accept or reject the Solution. If the Solution is rejected, an opponent must show that it violates at least one of the criteria in section VII-B. A Solution is correct if no opponent shows that it is incorrect.

In a three-player match, a player who does not present a Solution for a shake scores 6 if he accepts another player’s Solution as correct even if that Solution is subsequently proved wrong by the other checker.

Comment Players must not use the cubes in Required, Permitted, and Resources to form the Solution being checked since this causes arguments over where each cube was played.

1. A player who claims an opponent’s Solution does not equal the Goal must give at least one of the following reasons.
1. The Goal has no legal interpretation.

Examples

• 7 ÷ 0 when 0 is not wild
• A Goal containing more than six cubes or a three-digit number
• Mid/Jr/Sr: 39 in Base eight
• Elem: 3 √ 9
• Elem/Mid: 8 x when Sideways was not chosen
1. The Equation-writer’s interpretation of the Goal is not a legal interpretation.

Examples

• The writer makes 0 in the Goal equal another numeral when 0 wild was not chosen.
• The writer groups the Goal in an illegal manner;
• •• The Goal is grouped on the mat as 5x 3+4 and the writer interprets it as (5×3)+4.
• •• Elementary: with Smallest prime, the Goal x20x11 may not be interpreted x(20×11) since 20×11 is larger than 200.
• •• Senior: with √ = i, the Goal is 2√*47 and the writer interprets it as 2(√*47) (with no x after the 2).
• With Multiple operations, the Equation-writer uses an operation cube in the Goal multiple times.
• With red Exponent, the writer interprets the Goal 312 (red 2) asa three-digit
1. The Solution does not equal the Equation-writer’s interpretation of the Goal.
• Checkers must make an effort to determine whether the Solution equals the writer’s interpretation of the Goal before rejecting a Solution.
• The checker can give a general argument that the Solution does not equal the Goal.

Examples

• The Goal is a fraction or an irrational number, but the Solution equals an integer (or vice-versa).
• The Solution equals a value greater than 1000 when the Goal is 50×10.
1. One or both sides of the Equation may be grouped so that the Solution does not equal the Equation-writer’s interpretation of the Goal.

If an opponent believes there is a legal interpretation of a Solution or Goal which makes the Equation wrong, that opponent should copy the Solution and/or Goal to his own paper and add symbols of grouping where they will create a wrong interpretation. If the checker is able to do so, the Solution is incorrect. Examples

• The Solution (left side) in 5 * 2 – 4 + 0 = 3x(5+2) is ambiguous. An opponent may rewrite it as 5 * (2 – 4) + 0 so that it does not equal 3x(5+2).
• The Solution (left side) in 2 x 4 – (3 + 1) = 4 is ambiguous. An opponent may rewrite it as 2 x [4–(3+1)] so that it does not equal 4. However, an opponent may not rewrite it as 2 x [4–(3] + 1) since the brackets interfere with a grouping already in the Solution.
• The Goal (right side) of (6 x 4) – 2 = 7 + 5 x 3. A checker who rewrites the Goal as (7+5)x3 has shown that the Solution is incorrect.

Comment  Certain variations (such as 0 wild) allow cubes to be used for other symbols. If a cube stands for anything other than what is on the cube, the Equation-writer must indicate clearly and unambiguously in writing what each such cube represents. (See Appendix A for a list of suggested ways of doing this. Appendix A also lists the default interpretations when players do not write what symbols represent.)

1. A symbol or group of symbols is used ambiguously in the Solution or Goal, and one interpretation of the symbol(s) gives a value that makes the Solution not equal the writer’s Goal.

Example

Jr/Sr: With Base twelve, the Solution (left side) in 6 + √4 = 5 + 3 is ambiguous because √ can mean root or the digit eleven.

Note: See Appendix A for the default meaning of symbols that may be ambiguous. For example, if 0 is wild and the Equation-writer does not indicate what 0 means, 0 equals 0.

1. A variation is applied wrongly or not at all.

Examples of incorrect Solutions

• With 0 wild, the Solution uses a 0 for one symbol and another 0 (or a 0 in the Goal) for a different symbol.
• Elem/Mid: With Average, the Solution equals the Goal if + is interpreted as addition but not as average.
• Mid/Jr/Sr: With Multiple of k, the Solution equals the Goal rather than differing from it by a multiple of k.
• Last Cube Procedure
1. If one cube remains in Resources, the player whose turn it is must either move that cube to Required or Permitted or challenge Never. If he moves the cube, each player has two minutes to write a Solution.

The last cube in Resources may not be moved to Forbidden. If a player does so, any challenge that is made is set aside and the cube is returned to Resources. There is no penalty unless the player’s time to move expires. (See section XI.)

1. An opponent may challenge Never against the player who moved the last cube provided the challenge is made by the end of the first minute for writing Solutions. If the challenge is made, any Equation-writer has the rest of the original two minutes to write a Solution.

Comment Any Now challenge against the player moving the last cube is invalid as is any Never challenge made after the first minute for writing Solutions. In both cases, the player attempting to challenge loses a point and the challenge is set aside.

1. Illegal Procedures
1. Any action which violates a procedural rule is illegal procedure. A player charging illegal procedure must specify (within 15 seconds) the exact nature of the illegal procedure.
1. If a move is illegal procedure, the Mover must return any illegally moved cube(s) to their previous position (usually Resources) and, if necessary, make another move.

The Mover must be given at least 10 seconds to make this correction, unless the original move was made after the ten-second countdown (see section XI-A-3 below), in which case the time limit rule (section XI-A) is enforced. In general, there is no direct penalty except that the Mover may lose a point if he does not legally complete his turn during the time limit.

Examples of illegal procedures

Moving out of turn, moving two cubes without calling “Bonus” before the first cube touches the mat in Forbidden, moving the last cube in Resources to Forbidden

1. If the move is not illegal procedure, the cube stands as played.

Comment There is no penalty for erroneously charging illegal procedure. However, see section IX-C if a player does so frequently.

1. An illegal procedure is insulated by a legal action (for example, a move or challenge) by another player so that, if the illegal procedure is not corrected before another player takes a legitimate action, it stands as completed.

Example     Suppose the player in the lead makes a bonus move. Before anyone notices the illegal procedure, the next mover moves (or a valid challenge is issued). In this case, the illegal bonus move stays in Forbidden without penalty.

1. Certain forms of behavior interfere with play and annoy or intimidate opponents. This includes not playing to win but rather trying only to ruin the perfect scores of one or both opponents (for example, by erroneously challenging Now or Never at or near the beginning of each shake so that both opponents will score 5 for the round). If a player is guilty of such conduct, a judge will warn the player to discontinue the offensive behavior. Thereafter during that round or subsequent rounds, if the player again behaves in an offensive manner, the player may be penalized one point for each violation after the warning. Flagrant misconduct or continued misbehavior may cause the player’s disqualification for that round or all subsequent rounds. Judges may even decide to have the other two opponents replay one or more shakes or the entire round because play was so disrupted by the third party.

Examples This rule applies to constant talking, tapping on the table, humming or singing, loud or rude language, keeping a hand or finger over or next to the challenge block, making numerous accusations of illegal procedure that are not illegal procedure, challenging Now at or near the beginning of each shake when no Solution can be made with one more cube with the intent of making the opponents tie, and so on.

1. Scoring a Shake
1. After a challenge, a player is correct according to the following criteria.
1. That player had to write a Solution and did so correctly.

If the Third Party agrees with the person who must write a Solution, the Third Party must write a correct Solution also.

1. That player did not have to write a Solution (someone else did), and no opponent wrote a correct Solution.
1. After a challenge, points are awarded as follows.
1. Any player who is not correct scores 2.
2. Any player who is correct scores 6, unless that player is the Third Party agreeing with the Challenger, in which case the score is 4.
2. After the last cube from Resources is moved to the playing mat and no one challenges Never, points are awarded as follows.
1. Any player who writes a correct Solution scores 4.
2. Any player who does not write a correct Solution scores 2.
3. A player who is absent for a shake scores -4 for that shake.
1. Time Limits
1. Each task a player must complete has a specific time limit. The one- and two-minute time limits are enforced with the timer. If a player fails to meet a deadline, he loses one point and has one more minute to complete the task. If he is not finished at the end of this additional minute, another one-point penalty is imposed and he loses his turn or is not allowed to complete the task.

Note: In Elementary and Middle Divisions, each one-point penalty (for whatever reason) must be approved (initialed) by a judge on the scoresheet.

1. The time limits are as follows.
 a. rolling the cubes 1 minute b. making a variation selectionThis time limit does not begin until after the one minute for rolling the cubes. 15 seconds c. setting the Goal 2 minutes d. first turn of the player to the left of the Goal-setter 2 minutes e. all other regular turns (including any bonus moves) 1 minute f. stating a valid challenge after picking up the challenge block 15 seconds g. deciding whether to challenge Never when no more cubes 1 minute

remain in Resources

If the Never challenge is made, any time (up to a minute) the Challenger takes deciding to challenge counts as part of the two minutes for writing a Solution.

1. writing a Solution                                                                               2 minutes

During this time, the Third Party (if there is one) must decide whether to present a Solution after a Now or Never challenge.

1. deciding whether an opponent’s Solution is correct                   2 minutes
1. Often a player completes a task before the time limit expires. When sand remains in the timer from the previous time limit, the next player will receive additional time. An opponent timing the next player may either flip or not flip the timer so as to give the opponent the lesser amount of time before the remaining sand runs out and the next time limit can be started.
2. A player who does not complete a task before sand runs out for the time limit must be warned that time is up. An opponent must then count down 10 seconds. The one-point penalty for exceeding a time limit can be imposed only if the player does not complete the required task by the end of the countdown.

The countdown must be done at a reasonable pace; for example, “1010, 1009, …, zero.”

1. A round lasts a specified amount of time (usually 30 or 35 minutes). When that time is up, players are told not to start any more shakes. Either of two procedures may be used to determine the end of the shake in progress when time is called.
1. Players finish the last shake regardless of how much time it takes.

Comment This procedure can be used in local play when only one round is played on a given day.

1. Players have five minutes to finish the last shake. After these five minutes, players still involved in a shake in which no challenge has been made and one or more cubes remain in Resources will be told: “Stop; do not play another cube to the mat. Each player has two minutes to write a correct Solution that may use any of the cubes remaining in Resources.” Any player who presents a correct Solution scores 4 points for that shake; a player who does not present a correct Solution scores 2.
• Scoring a Match
1. Each player is awarded points for the match based on the sum of his scores for the shakes played during that match according to the following tables.
 Three-Player Matches Points first place 6 two-way tie for first 5 three-way tie for first 4 second place 4 tie for second 3 third place 2
 Two-Player Matches Points first place 6 two-way tie for first 5 second place 4 did not play 0
1. When a round ends, each player must sign (or initial) the scoresheet and the winner (or one of those tied for first) turns it in. If a player signs or initials a scoresheet on which his score is listed incorrectly and there is evidence that there was intent to deceive and the error was not a simple oversight, then do the following.
1. If the error gives the player a lower score, he receives the lower score.
2. If the error gives the player a higher score, he receives 0 for that round.
1. Elementary Variations (grade 6 and below)

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.

Examples

• 3 * 2 is acceptable and equals 9. 0 * 9 equals 0 and 7 * 0 equals 1. However, 2*(1–3), 4*(1÷2), (2–5)*4, and (2÷3)*3 are not legal in Elementary.
• 2√9 or just √9 is acceptable and equals 3. 9√0 equals 0. However, √5 and 3√9 are not legal since neither is a whole number. Also 2√(1÷3), (1÷2)√5, and 3√(1–9) are illegal in Elementary.
• The legality of √3*4 depends on its grouping. √(3*4) is legal; (√3)*4 is not.
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.

Examples 1 + 2 +  = 1 + 2 + .5 = 3.5; 1 ÷  = 1 ÷ (1/3) = 1 x 3 = 3

Comment See Appendix A for acceptable ways to indicate a sideways cube in a Solution.

1. 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.

Examples 6 x  = 6 x (–2) = –12. However, 6  is not legal for 6 – 2 or 60 + (–2).

Comment See Appendix A for acceptable ways to indicate an upside-down cube in a Solution.

1. 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 Equationwriter must specify in writing the interpretation of the 0 cube if it stands for anything other than 0 in the Solution.

Examples

• (0 x 6) – 0 = 15, where both 0’s stand for 3, is allowed but (0 x 6) – 0 = 14, where the first 0 stands for 3 and the second for 4, is not
• (0 x 6) – 0 = 12, where the first 0 stands for 2 and the second for 0, is not
• A 0 in the Goal and any 0 in the Solution must equal the same number. So (8 x 5) + 0 equals the Goal 40 if each 0 equals 2. However, (9 x 5) – 0, where this 0 stands for 5, does not equal the Goal 40.
1. 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! (“5 factorial”) means 5 x 4 x 3 x 2 x 1, which equals 120. n! is defined only for whole number values of n. 0! is defined as 1.
• In the absence of grouping symbols, ! applies to just the numeral in front of it. Examples 4 + 7! means 4 + (7!). An opponent may not interpret it as (4 + 7)!

3√5! means 3√(5!), not (3√5)!

Suppose the Goal is 4 + 7. If an Equation-writer wants 4+(7!), just write 4+7! If the writer wants 11!, write (4+7)!

With Number of factors, 4+x7! means 4+x(7!), not 4+(x7)!

Examples

• For the Goal 4 x 30, a Solution of 5! is not correct since it contains only one cube.
• If the Goal is 9 ÷ 8, an Equation-writer may interpret it as 9! ÷ 8!, which is 9. However, the Solution may not contain an ! since both allotted factorial signs have been placed in the Goal (unless Multiple operations has been called – see 10 below).
• The Equation 5 x 4 ÷ 0! = 20 is correct since 0! equals 1.
• The Equation (8 – 5)! + 2 = 4! ÷ 3 is correct since the Goal is (4 x 3 x 2) ÷ 3 = 4 x 2 = 8. (e) 3!! = (3!)! = (3 x 2)! = 6! = 6 x 5 x 4 x 3 x 2 = 720

The following even-year variations will be used in 2007-8.

1. Two-digit numerals Two-digit numerals are allowed in Solutions.
2. Three-operation Solution Any Solution must contain at least three operation symbols. The operation symbols are +, –, x, ÷, *, and √ .

Comment If the Factorial variation is also chosen for the shake, each ! sign in a Solution counts as an operation symbol. x used for number of factors is an operation symbol. So is an upside-down radical used for percent. However, a * used as a decimal point does not count as an operation symbol.

1. Smallest prime xA means “the smallest prime bigger than A,” where A is a rational number ≤ 200.

Comment This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube will usually be clear from the context since smallest prime is a unary operation and multiplication is a binary operation. For example, the Goal 4xx6 has only one interpretation: 4 x (x6), which is 4 x 7 = 28. Examples

• x7 = 11, the smallest prime bigger than 7.
• x(9 ÷ 2) = 5, the smallest prime bigger than 4.5.
• x(0 – 3) = 2, the smallest prime bigger than –3. (Note: 1 is not prime.) (d) xx5 = x(x5) = x7 = 11.
• In the expression 2xx5, the first x means multiplication and the second means smallest prime. The value of this expression is 2 x (x5) = 2 x 7 = 14.
• In the expression x5x7, the first x means smallest prime and the second means multiplication. The value of this expression is either x(5×7) = 37, the smallest prime bigger than 35, or (x5)x7 = 7 x 7 = 49.
• There is no limit to the number of consecutive x’s in an expression, especially with the Multiple operations variation also in effect. Thus xxx9 = xx(x9) = x(x11) = x13 = 17.
1. 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 twodigit numeral.

Examples

• In the Goal or Solution, 25 16 = 25% of 16 = 0.25 x 16 = 4.
• In the Goal or Solution, 6 8 = 6% of 8 = 0.06 x 8 = 0.48.
• In a Solution (8 – 3)  (4 + 2) = 5% of 6 = 0.05 x 6 = 0.3.
• If the Decimal point variation (see below) is also in effect, an expression like 1.5 .25 is legitimate in a Solution and equals 1.5% of 0.25 = 0.015 x 0.25 = 0.00375. Similarly, in a Solution, 25 (1.5 x 2) = 25% of 3 = 0.75. And 12.5            16 = 2.
1. 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.

Comment This variation does not rule out using * for exponentiation. Therefore, Equation-writers are encouraged to write a decimal point instead of * when they want to use * as a decimal point. Also one * may be used as a decimal point and another for exponentiation. If * is used as a decimal point, this must be indicated in writing in the Solution. (See Appendix A.) Examples

• 2*5 means either 2.5 or 25; 3*0 means either 3.0 or 30. The Equation-writer must indicate whether the decimal interpretation is desired. One way to do this is to write 2.5 or 3.0 rather than 2*5 or 3*0.
• 2*4×2 = 2.4 x 2 or 28 or 24 x 2; it does not mean 2.8 [that is, it may not be grouped as 2.(4×2)].

Similarly 4*3! may not be interpreted as 4.(3!) or 4.6. (4*3! has no defined interpretation.)

• 12*5 means either 12.5 or (in the Goal only unless the Two-digit numerals variation has been chosen) 125; 1*25 means 1.25 or (in the Goal only unless the Two-digit numerals variation has been chosen) 125.
• 15*0 means either 15.0 or (in the Goal only unless the Two-digit numerals variation has been chosen) 150.
• In the Goal or Solution, 255* means 255 and *050 means .05.
• 15*25 = 1525 (in the Goal only unless the Two-digit numerals variation has been chosen) but has no legitimate interpretation as a decimal.
• 122*5, 1*225, *1225, and 1225* have no interpretations as decimals or powers.
• The expression *37*5 may not be interpreted as .37 1/2 and has no defined interpretation in Elementary Division.
• The “digits” are the symbols 0,1,2,3,4,5,6,7,8, and 9. A digit turned sideways or upside-down is no longer a digit. Therefore, with Sideways or Upside-down in effect, you may not use an expression like * , *, * , or * and interpret the * as a decimal point. The following odd-year variations will be used again in 2008-9.
1. 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.

• After a Never challenge, any operation sign in Resources may be used many times in a Solution. After a Now challenge, if the one cube allowed from Resources is an operation sign, it may be used multiple times.
• Players may simply write an operation sign multiple times in Solutions without any additional comment since an operation cube is not being used to represent another symbol.
1. Average + shall not represent addition; instead it shall represent the operation of averaging two

Examples

• 7 + 9 = 8, the average of 7 and 9. 7 – (0 – 9) equals 16, as usual.
• 5 + (4 x 2) = the average of 5 and 8 = 6.5.
• The Goal 4+6+9 has two values: (4+6)+9 = 5+9 = 7; 4+(6+9) = 4+7.5 = 5.75. Notice that neither answer equals 19/3, the usual (mathematical) average of 4, 6, and 9.
1. LCM √ may represent the LCM (least common multiple) of two counting numbers.

Comment This variation does not rule out using √ for root. So each Equation-writer must indicate in writing which √ in the Solution represents LCM. (See Appendix A.) Examples

(a) 6 √ 8 = 24.                                           (b) (2 x 3) √ (5 + 4) = 6 √ 9 = 18.

(c) 2 √ 4 = 4 (or 2, the square root of 4). (d) 0 √ 5 is undefined.

(e) 6√√9 means the LCM of 6 and the square root of 9; that is, the LCM of 6 and 3, which is 3.

1. GCF * may represent the GCF (greatest common factor) of two whole numbers, provided at least one of them is not 0.

GCF(A, B), “the greatest common factor of A and B,” is defined if A and B are counting numbers or if A is a counting number and B = 0 or if B is a counting number and A = 0. GCF(A, 0) = A and GCF(0, B) = B.

Comment This variation does not rule out using * for exponentiation. So each Equation-writer must indicate in writing which * in the Solution represents GCF. (See Appendix A.) Examples

(a) 8 * 6 = 2 (or 86).                                   (b) (4 x 2) * (6 + 3) = 1 (or 89).

1. Number of factors xA means “the number of counting number factors of A,” where A is a counting number less than or equal to 200.

Comment This variation does not rule out using x for multiplication. In the Goal or Solution, the meaning of an x cube will usually be clear from the context since number of factors is a unary operation and multiplication is a binary operation. Examples

• x(6 x 2) = 6 (since 12 has six factors: 1, 2, 3, 4, 6, 12)
• x(4 x 4) = 5 (since the factors of 16 are 1, 2, 4, 8, 16)
• x12 = 6 (for use in any Goal or, if the Two-digit numerals variation is chosen, in a Solution) (d) x0, x(1 ÷ 2), x(1 – 4), and x(5 * 4) are not defined.
• xx12 = x(x12) = x6 = 4
• In the expression 3xx7, the first x means multiplication and the second means number of factors. 3xx7 = 3 x (x7) = 3 x 2 = 6.
• In the expression x4x2, the first x means number of factors and the second means multiplication. The value of this expression is either x(4×2) = 4 (the number of factors of 8) or (x4)x2 = 3 x 2 = 6.
1. Middle Division Variations (grade 8 and below)

The following four Elementary variations are also played in Middle Division. (See the comments and examples after each variation in the Elementary list.)

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 his Solution.

Example     If a player interprets 0 in the Goal as x, then any 0 in that player’s Solution must also be an x.

• If 0 wild and Multiple operations are both chosen, 0 may be used multiple times in a Solution only if it stands for an operation sign, not a numeral.
• If 0 wild and Factorial are both in effect, 0 may not stand for ! because ! is not a symbol on the cubes.
• If Base eight is also chosen (see below), 0 may not represent the digits “8” or “9.” If Base nine is chosen, 0 may not represent “9.”
• With Number of factors (see below), one 0 may mean number of factors and another 0 may be multiplication since 0 is the symbol x in both cases.
1. 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.

The following variations may also be chosen every year in Middle Division.

1. 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.

Examples

• For Base eight, 37 + 5 = 6 * 2 is a correct Solution. Any Solution or Goal containing the digit “8” or “9” is an illegal expression.
• For Base nine, 34 + 5 = 6 * 2 is a correct Solution. Any Solution or Goal containing the digit “9” is an illegal expression.
• In Base eight, a Goal like 3 + 8 or 39 should be challenged Never. A Goal like 39 is also illegal in Base nine.
1. 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.

Example If k = 6 and the Goal is 5, then a Solution must equal 11, 17, 23, 29, and so on, or –1, –7, –13, –19, and so on. A Solution equal to 5 is incorrect.

Comment Multiple of k does not require any special writing of the Goal by an Equation-writer. As always, write the interpretation of the Goal, indicating wild cubes and grouping. You do not have to indicate the multiple of k difference.

Note           k = 12 is no longer an option.

These two Elementary even-year variations will be played in Middle in 2007-8. (See the comments and examples after each variation in the Elementary list in addition to any comments and examples below.)

1. Percent 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.

If Base m and Percent are both chosen, the meaning of percent (“per 100”) changes with the base. “Percent” means “per sixty-four” for Base eight and “per eighty-one” for Base nine.

Example     In Base eight, 60 11 = (60eight ÷ 100eight) x 11eight = (48ten ÷ 64ten) x 9ten = (3ten ÷ 4ten) x 9ten = 27ten ÷ 4ten = (6 + 3/4)ten = (6 + 6/8)ten = 6.6eight

1. 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.

Example      A Goal of 4**5 can equal either 4.5 = 1028 or 4.5 = √4 = 2.

Comment If 0 wild and Decimal point are both chosen, 0 may represent a decimal point. Also, one 0 in the Solution may be decimal point and another 0 may be exponentiation since 0 is the same symbol, *, in both cases.

The following two Middle even-year variations will be played in 2007-8.

1. 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
 Examples In the Goal or Solution, the expression 45+ means either .454545… or .455555… The first interpretation, .454545…, equals the fraction 45/99, which reduces to 5/11. The second interpretation equals (45-4)/90 = 41/90. 33+ means .3333… under either interpretation. Comment If the Equation-writer does not indicate which interpretation of AB+ in the Goal or So-

lution he is using, the default interpretation is .ABABAB… (See Appendix A for all matters involving how Solutions are written.)

1. 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.

Examples

• If the chosen color is red, the Goal 253, where the 3 is red, must means 253 since three-digit numerals are illegal.
• If blue is the chosen color, a Solution like 52, where the 2 is on a blue cube, is legal.

• The exponent of the selected color applies to just the numeral in front of it unless grouping symbols are used.

Examples 4+32 means 4 + (32). An opponent may not interpret it as (4+3)2.

√5! 2 means √(5! 2), not (√5!)2.

With Number of factors, 4+x122 means 4+x(122), not 4+(x12)2.

With Factorial, 4+3!2 means 4+(3!)2.

• If Factorial is also chosen, a ! may be placed behind an exponent. So with red exponent, a Goal of 23 (red 3) may be interpreted as 23! or 26.
• If a player selects Exponent and no digits of the selected color were rolled, that player is penalized one point and must pick another variation.

The following three Elementary odd-year variations will be used again in Middle in 2008-9. (See the comments and examples after each variation in the Elementary list in addition to any comments below.)

1. 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.
2. Average + shall not represent addition; instead it shall represent the operation of averaging two

Example     If 0 wild is chosen along with Average, any 0 that represents + must mean average.

1. Number of factors xA means “the number of counting number factors of A,” where A is a counting number.

Example     If 0 wild is chosen along with Number of factors, one 0 may represent number of factors while another 0 may be multiplication since 0 is the symbol x in both cases.

Note           The requirement that A be less than 1000 has been removed for Middle Division.

The following Middle odd-year variation will be used again in 2008-9.

1. 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.

Examples

• For Base ten, 9 + 1 may be interpreted as 9 + 1 (since 100 = 1), 9 + 10, 9 + 100, 9 + 1000, and so on, or as 9 + 0.1 (since 10–1 = 0.1), 9 + 0.01, 9 + 0.001, and so on.
• If Base eight is chosen, then 1 may represent one, eight, sixty-four, and so on, or one-eighth, one-sixty-fourth, and so on. For Base nine, 1 represents one, nine, eighty-one, one-ninth, etc.
1. Junior Division Variations (grade 10 and below)

The following two variations are in effect for all shakes in Junior and Senior Division. (See the examples and comments for each variation in the Elementary list.)

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.

The following Middle variations may also be chosen in Junior Division. (See the comments and examples after each variation in the Middle or Elementary list in addition to any comments below.)

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 his Solution. The player selecting this variation specifies whether 0 or x (but not both) is wild for the shake.

Examples for x wild (for 0 wild, see the examples for Elementary Division on page E13 and for Middle on page E16)

• x – (x ÷ 3) = 4, where both x’s stand for 6, is a correct Solution. (b) (9 x 3) x 5 = 1, where both x’s stand for –, is a correct Solution.
• x – (3 x 2) = 2, where the first x is 7 and the second x is +, is not a correct Solution.
• An x in the Goal and any x in the Solution must represent the same symbol. For example, the Solution of (4 * 2) – 2 = 2×7 is incorrect since x stands for * on the left side and (by default) x

↑ x

stands for multiplication in writer’s Goal.

1. 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.
2. 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.

If Sideways and Base eleven (or twelve) are both chosen, an * may be used sideways to represent one-tenth. If the * is part of a two-digit numeral, it may not be interpreted as sideways. If an * is a one-digit numeral in the Goal, the Equation-writer may interpret the * as right-side up or sideways regardless of the way the * is physically placed in the Goal. In a Solution, the writer must clearly indicate if an * is sideways.

Note            Base ten is no longer a choice in Junior/Senior Division.

• In bases eleven and twelve, * may still represent exponentiation; in base twelve, √ may still represent root. If the interpretation of an * or √ is not clear from the context of the Solution, the Equation-writer must indicate which meaning is desired so as to eliminate any ambiguous in the Solution or the Goal. (See Appendix A.)
• If Powers of the base is chosen with Base eleven, 1 may mean one, eleven, one-hundred twenty-one, and so on, or one-eleventh, one one-hundred twenty-first, and so on. If Powers of the base is chosen along with Base twelve, 1 may mean one, twelve, one-hundred forty-four, and so on, or one-twelfth, one one-hundred-forty-fourth, and so on.
• If 0 (or x) wild is chosen along with Base eleven or twelve, a wild cube may represent * for ten or √ for eleven (or exponentiation or root as long as each wild cube represents the same symbol).
1. 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.

Note           k = 12 is no longer an option.

1. 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.
2. 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 ! must be in writing. However, if Multiple of k is also chosen for the shake, no factorial may be placed in the Goal.
3. Number of factors xA means “the number of counting number factors of A,” where A is a counting number.

Comment Since there is no limit to the size of A, it is possible to present a Solution that is uncheckable. For example, with Multiple of k =11 and Factorial: x(8!! + 1)  =  5

Any such Solution that cannot be verified (even with a calculator) by opponents and judges as correct or incorrect will be ruled incorrect.

1. 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.

Note           This variation, formerly an even year variation, will now be played every year in Junior/Senior.

Note: The AB+ and Average variations are no longer played in Junior/Senior Division.

1. Senior Division Variations (grade 12 and below)

Players may choose any of the Junior variations (except for the two which are in effect for every shake) plus the following.

shall not represent the root operation but instead may represent the

imaginary number i (such that i 2 = –1). The √ may be placed immediately before or after a numeral without the x sign.

With this variation, the rules for legal expressions in section III-C are amended to allow expressions like a * (b ÷ c) where a is a negative real number, b is an integer, and c is an even non-zero integer (when b ÷ c is reduced to lowest terms). Furthermore, in a Goal or Solution, any expression of the form a * (b ÷ c) (where c ≠ 0) may equal any one of the complex roots equal to the expression. An Equation-writer using such an expression must indicate in writing which one of the complex roots the expression equals. [See example (f) and comment (d) below.] Examples

• 2i may be represented in a Goal or Solution by either 2√ or √2. √ or √ is .25i. √ or √ is –3i.
• 3 + 4i may be represented as either 3 + (4√) or 3 + (√4). (c) (3+4)√ or √(3+4) equals 7i.
• i 6 may be represented as √ * 6.
• 14i may be represented as 7√ 2.
• A Goal of 4 * (1 ÷ 2) may equal 2 or –2. A Goal of 16 * may equal 2, –2, 2i, or –2i. Each Equation-writer must eliminate any such ambiguities in his Solution and interpretation of the Goal.
• An expression like 4*√ is not allowed because the exponent is not a real number. Comments
• “Numeral” means “any expression that names a number, real or otherwise.” i itself is a numeral, which means that expressions like √√, √√√, and so on, are legal and equal i2, i3, and so on.
• The variation says √ may represent the imaginary number i, instead of must represent, only because Base twelve may be chosen. In this case, √ may also equal the digit eleven.
• If 0 (or x) wild is also chosen, any wild cube used as √ must represent i. (Exception: Base twelve)
• Suppose the Goal is 0 – 8 √ . Then a Solution might be this: (8 x 2) * (3 ÷ 4). The Equationwriter must indicate in a clear and unambiguous manner which root is being used. One way is this:                                               (8 x 2) * (3 ÷ 4)

(2i)3

• With √ = i, the Goal and the Solution may equal non-real (complex) numbers.
• If Multiple operations is also chosen, √ may not be used multiple times because it does not represent an operation when √ = i is in force.
• A Goal like 2√*88 may not be interpreted as 2(√*88) since the variation allows a numeral in front of √ without x but not in front of an expression like √*88 without a x sign. Similarly, the expression 2√*8 in the Solution must be interpreted as (2√)*8 and not 2(√*8) (whether the Equation-writer includes the parentheses around 2√ or not).
1. Decimal in Goal Each Equation-writer may determine where decimal points occur in the Goal.

Examples

• A Goal of 20 may be interpreted as 20, 2.0, or .2.
• A Goal of 2 * 3 may be 2 * 3, .2 * 3, 2 * .3, or .2 * .3.

Comment A decimal point may be placed in front of only a right-side up digit. Therefore, no de-

cimal point may be placed in front of a sideways or upside-down cube or in front of i.

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

Examples

• [(6 x 4) + 1] 5 = log525 = 2.
• 3 2 = log23, which is an irrational number.
• a  1 is undefined for any value of a. 0  5, (0 –1)  1, (0 – 8)     (3 – 1), and 4     (0 – 2) are all undefined.

We recommend checking out https://schoolyourself.org which offers interactive math lessons.

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