Appendix A – Ways of Indicating What Cubes Mean in Equations Some General Principles

  1. Each Equation-writer must not only create a correct Solution but must also clearly communicate the Solution to the Checker(s) so that they can verify that the Solution equals the writer’s interpretation of the Goal.
  2. A Equation-writer must remove all ambiguity from the Solution and Goal so that there is no question that the two sides of the Equation are equal. Removing ambiguity has two components: (a) using grouping symbols to specify the order of operations and (b) indicating the value of any cube that may have multiple meanings. It is component (b) that is the subject of this Appendix, although in some cases placement of parentheses may clarify the meaning of a symbol.
  3. In general, a Equation-writer should write in the main line of the Equation the value of each cube that represents something other than its “face value.” For example, write the value of the wild cube in the Equation and indicate above or below that the value comes from a wild cube. This principle is implemented in the Recommended methods in this Appendix. The reverse technique, listing the wild cube in the Equation and indicating its value from the side, is Acceptable
  4. If the Equation-writer does a good job, the Checker(s) should not have to ask a single question about the Equation. It should be clear what each symbol means and which interpretation of the Goal the writer has chosen.
  5. In general, arrows are preferable for indicating what a cube means, like this.



The arrow can come from above or below and can point to or from the symbol in the Equation. Writing the meaning just above or just below the mainline of the Equation without an arrow is acceptable but has the drawback that the two digits may overlap and confuse rather than clarify. This should not be the case where several letters like sw or ud indicate the meaning; hence an arrow is not needed (although acceptable) in these situations.

Explanation of Terms

Methods of clarifying symbols in Equations are divided into three categories in the list in this Appendix: Recommended, Acceptable, and Unacceptable. Here are the intended meanings of these terms.

Recommended     This is the method that should be taught to players.

Acceptable             Any method in this category will be accepted by judges as correct.

Unacceptable            These methods will cause the Equation to be ruled incorrect by judges.  This Appendix also lists a Default interpretation for many ambiguous symbols. The Default meaning is how the symbol will be interpreted if the writer does not clarify its meaning. If a symbol has no Default meaning, the writer must indicate the desired interpretation. Otherwise, the opponent may interpret it so as to make the Solution incorrect.


Last update 8/15/07

I.  Equations

  1. Correct form for writing Equations Recommended Solution = Goal

Acceptable             Goal = Solution

Solution =                           or                     Goal =

Goal                                                             Solution

Unacceptable         Solution  [that is, no Goal written]

  1. Ways of writing what cubes mean in Solutions and Goals.
Division Variations Examples Default
All Sideways Recommended:  7swswc      sw  sc ↓Acceptable:    ,  7 ,  7,  7,   7,  7, 7, etc.

sc  swc             ↑          sw

Unacceptable: 1/7, 1÷7, etc. (that is, no division symbol of any kind)

Cube is right-side up.
All Upsidedown Recommended:  7ud usd              ud ↓Acceptable: ,  7 ,  7,  7, 7, etc.usd       ↑          ud

Unacceptable: –7, (–7), 7 (that is, no subtraction or negative symbol)

Cube is right-side up.
All 0 wild (with0 used as adigit)                                           0↓Recommended:  2 x (8 + 7)      or2 x (8 + 7)

2           0         2

↓                   ↑

Acceptable:  0 x (8 + 7),  0 x (8 + 7), 0 x (8 + 7),  0 x (8 + 7), 2 x (8 + 7) etc.

↑                    ↓                   0

2                          2

(Also see MJS 0 wild section below.)

0 = zero (if placed so that it must be a digit). Note: It is sufficient to indicate what one 0 represents; it is understood all other 0’s (Goal and Solution) equal the same symbol.
All Sideways, 0 wild                                           0↓Recommended:  7                 sw0 sw 7  sw 0

↓ ↓ ↓ ↓

Acceptable:       , 0,  0,  7,      , 0,   0,  7

↑ ↑ ↑ ↑

0 sw 7  sw 0

Cube is right-side up and 0 = zero.


Division Variations Examples Default
or the same methods with the arrows pointing the opposite way (or no arrows at all) or sc or swc in place of sw, etc.
All Upsidedown, 0 wild                                            0↓Recommended:  7                ud0 ud 7  ud 0

↓ ↓ ↓ ↓                                     ud

Acceptable: ,  0,  0,  7, , 0,   0, 7, 7

↑ ↑ ↑ ↑ 0

0 ud7              ud0   or same methods with the arrows pointing the opposite way (or no arrows at all), uc or usd in place of ud, etc.

A 0 in the Goal is ambiguous for upsidedown. Default is 0 cube is right-side up zero.
All Multiple operations Recommended:    if used consecutively (or number below or no arrow). In any case, you may write the operation sign as many times as you want it. The operation is used just once.
All # factors or(E only) smallest prime 6 x 7 → must be multiplication x67 → # factors (or smallest prime) 8xx7 or 8x(x7) → 1st x = mult., 2nd = # factor (or sm. prime)xx5 or x(x5) → both # fac. (or sm. prime) Context (placement of the symbol) determines the interpretation of the x. Usually no indication is necessary.
All Mult. op., # factors or (E only) sm. pr. Recommended: x2    or   x210 ↵              10 x’s↵ (or point from top) The x is used just once.
E LCM Recommended: 8 √ 2   or      LCM↑                        ↓LCM                8 √ 2Acceptable: Same with no arrow. √ = root
E GCF Recommended: 8 * 2   or      GCF↑                         ↓GCF                8 * 2Acceptable: Same with no arrow. * = exponentiation
EM Decimal point Recommended: .23, 23.+5, 2.3 (that is, write a decimal point, not *)Acceptable: *23, *23,  *23 (or samedp ↑ ↑     . decmethods with indication from above) In most cases, context determines what * means; for example:

*23 → decimal point

23*+5 → decimal point (continue next page)

If context (placement of the symbol) does not determine, default to exponentiation. For order of operations, decimal point takes precedence. (See the examples listed after the variation in Elementary variations section of the Tournament Rules.)


Division Variations Examples Default
2*3 → default to power unless player writes 2.3 or uses other methods above. 4**5 must be 4.5 in Elem. but is ambiguous in Middle.*125 can mean only .125.
EM Percent Recommended: 50%34 or 50 √ 34↑% or udAcceptable: 50  34 or second method above but point from above or no arrow

(or any acceptable method for upside-down cube – see page A1).

√ is right-side up (that is, root).
M Decimal point, AB+ AB+ cannot appear in a decimal expression (it has its own built-in decimal point).66+*5 is ambiguous; so default to 66+5. * = exponentiation
MJS 0 wild(0 as opera-tion) Recommended: 7+3 or     0↑ ↓           0 7*3Acceptable: Same as above with 0 and symbol interchanged and/or no arrow. If 0 is placed so that it must be a digit, it defaults to 0. There is no default meaning for 0 as an operation.
MJS Exponent Recommended: 52, (4+1)2,  52,   exp↑ ↓*  52Acceptable: Same as last two above with no arrow or with “re” for red exponent, etc. Unacceptable: 5*2 or 5^2 (that is, no exponent sign of any kind) Two consecutive digits (with the second the exponent color) in the Goal and (JS only) in Solutionwith base m defaults to a two-digit number (no exponent). Goal of 723 (red 3 with red exp.) defaults to 723 (in Solution also with base m).
MJS 0 wild, red exponent Recommended: For a Goal of 703 with red 3, write: 703, 753, 7×3, 7+3, etc.↑              0Acceptable: Any method for writing 0 wild as an operation in the Equation (see p.A3). 703 defaults to 703. If 3 cannot be an exponent, there is no default.
MJS Powers of the base               1              ↓Recommended: 103, 100,            , etc.↑           1Acceptable: 1,   1 (or indicate from above)

↑ ↑

100 102

1 = one


Division Variations Examples Default
MJS AB+ For meaning of +, context dictates. For example, 45+5 → must be addition 45++5 → first + is rep. dec., 2nd is add. For indicating interpretation of AB+:Recommended: 45++5↑45/99 (or 41/90 or point from above) Acceptable: 45++5

__  ↑

.454545…, .45, .ABAB… or .ABB… (with/without dec. pt. and 3 dots), AB/99, (AB-A)/90 (or point from above or no arrow)

Unacceptable: 45+ ← AB (not clear; both  interpretations start AB)

45+ ← AB–A (just numerator of fraction)

Context dictates interpretation of +; AB+ defaults to the first interpretation(.ABAB…).
JS Base 11 or 12 Use of * and √ as digits creates ambiguities. If context does not determine meaning, player must indicate. Examples:7+*4 or 4*+7 → * is ten.3√4 is not ambiguous (without i in Sr.). 6**2 is ambiguous. In base 12, √4+2 is ambiguous.Recommended: (6*)*2 which means (6*)2 or 6*(*2) for 6*2 or 6 * * 2

exp.    ten

For √4+2, write (√4)+2 or (√4)+2 or

↑                ↑

root      11 or eleven

√(4+2) [√ must be root if no i in Senior].

Context determines; if context cannot determine, expression is ambiguous.
JS Base m,Powers of Base Side indications of pob may be in either base ten or base m as long as they are all in one base or the other.Recommended: (100 ÷ 4)  – (10 + 10)↑ ↑ ↑  1 1 1Acceptable:  With base 8,

                       (1 ÷ 4)  – (1 + 1)     or

↑                 ↑ ↑

100            10    10

                       (1 ÷ 4)  – (1 + 1)

↑                 ↑ ↑

64 (or 82)         8      8

(Continue on next page.)

1 Senior Contin= 1
Division Variations Examples Default
Unacceptable:  With base 8,                        (1 ÷ 4)  – (1 + 1)↑                 ↑ ↑100 or 102         8      8


Senior Division Variations are on the next page.

Division Variations Examples Default
S x wild See examples for 0 wild. x = x if placement of the x allows it to be an operation; also see the note for 0 wild for all divisions on page A1.
S √ = i Recommended: Since √ must be i, just write the √.Acceptable: Write i in place of √. None since (without base 12) √ must be i.
S 0 or x wild, √=i For 0 representing i in the Equation, use any method listed for 0 wild on page A3.To indicate 0 = i in the Goal 804:Recommended: 8√4Acceptable: 8i4 plus the methods for 0 wild in Equations

Unacceptable: 32i or 32√

S √=i, base 12  Placement of √ may determine its meaning. For example, in 3√4, 64√, or √64, √ must be i. However, 6+(2√), √+7, and 6√√2 are ambiguous.Recommended: If √ = i, write 6+(2i), i + 7,or 6ii2. If √ = 11, write 6+(2√), √+7, or 6√√2↑ ↑                         ↑ ↑

11 eleven eleven (or point from above)

Acceptable: 6+(2√)

i (or point from above or no ↑)

Context; if context does not determine, √ is ambiguous.
S Log Recommended: 8  2Acceptable: 8 ÷ 2, 8 ÷ 2, 8 log 2, log28↑ ↑log sw (or any other sw method)

Unacceptable: 8 ÷ 2 (÷ defaults to division)

÷ = division
S Decimal in Goal Recommended: Write a decimal point to indicate placement. For example, if the Goal is 15×8, write 1.5×8, 15x.8, .15x.8, etc.Acceptable: 15×8, 15×8, 15×8, etc.↑                 ↑ ↑ ↑.              dp

(Notice how carefully the arrow must be drawn to the exact place where the point goes.)

No decimal point in the Goal


II.  On-Sets

The following is the list of accepted ways of indicating in writing what cubes mean in OnSets Solutions.

Division Variations Examples Default
All Wild cube Recommended: R U G  or  B – V↑                      ↓B                UThe arrows may come from the top or not appear at all.

Acceptable: R B G  or  B U V

↑ ↓         U

The arrows may come from the top or not appear at all.

Wild cube = itself; it is also sufficient to indicate in one place in the Solution what the wild cube represents – it is understood to represent the same symbol throughout (Restriction and Set-Name).
All  inter-changeable Recommended: Write the symbol (U or ∩) you want in each place in the Solution; no other indication is necessary.Acceptable: Indicate the cube upside down in the same way as for a wild cube (above) or upside down cube in Equations.Caution: Writing the bar (_) on an upside-down symbol can cause confusion. It is best not to write the upsidedown symbols like this:  (for upsidedown intersect) or  (for upside-down union). U = U, ∩ = ∩
All V, /\ inter-changeable Same as for U, ∩ interchangeable with the same caution. V = V, /\ = /\
All Multiple operations Recommended: Write the operation sign as many times as you want; no special indication of mult. op. is necessary.Acceptable: For multiple consecutive primes:  R’↑                 4
JS Blank card wild Recommended: Draw a “picture” of the blank card with the colors indicated, like this:


Acceptable: Blank = BG

Blank card stays blank.