Jootsy Calculus

"In creative problem solving, it is frequently more important to look at a problem from different vantage points rather than run with the first solution that pops into your head." -- Eugene Raudsepp (Creative Growth Games)

in which 2 + 2 = 5,

among other things

translated by Michael Joseph Halm

Jootsy calculus is that branch of metamathematics in which calculations are not limited to just one alphanumeric system but allowed to JOOTS, "Jump Out Of The System" in which it began or had been previously "jootsed". The name is alternatively said to derive from that ancient practisioner, the Chinese philosopher and that protopataphysicist, Dzu-tse (also known by the Latinized Jucius and the Anglicized Jootsy).

With jootsy calculus, for example, two plus two not only can equal five, but an infinite number of other correct answers, depending on the viewer's pov ["point of view"] even including infinities. One, in fact, can use any mathematical representation included in the set of Generalized Orthographic Denotations.

"All things are possible with GOD."

One can add two letters to two letters to get a number word in any number of languages, two letters to two letters to get a Roman numeral, two multiplicative factors to get a quotient, use a different base for the

number system than the usual decimal, including "improper" representations using greater-than-base, half, imaginary or even irrational place holders, modulo arithmetic that notes only remainders.

For example here are a few of the possible answers to what

0 (ze + ro, Nim-addition)

1 (French [on + ze], German [ei + ns], Yiddish [ey + ns], Italian [on + ce], Japanese [ic + hí , ka + ta], modulos 3, 3/2, Swahili [mo + ja], Russian [od + in, od + na])

2 (French [de + ux], German [zw + ei], Italian [do + ce], Portuguese [do + is])

3 (French and Spanish [tr + es], German [dr + ei], Swahili [ta + tu], Yiddish [dr + ay, fir])

4 (bases 4+, Swedish [fy + ra], English [fo + ur], German [vi + er], Norwegian [fi + re], Roman [ij + ij])

5 (English [fi + ve], French [ci + nq], Swahili [ta + no], Swedish [fy + ra], Yiddish [fi + nf])

6 (Spanish [cuatro, se + is], Roman [ij + iv], French [quatre], German [se + is], Italian [si + ta], Norwegian [se + ks], Yiddish [ze + ks]

7 [French [se + pt], Italian [quattro], Portuguese [se + te], Spanish (sa + ba, Yiddish [zi + bn])

8, [French [hu + it], na + ne, German [oc + ht], Italian [ot + to], Latin [quattuor], Portuguese [oi + to], Roman [vi + ij], Spanish [oc + ho)])

9 [English [ni + ne], French [ne + uf], German [ne + un], (Japanese [ky + uu], Italian and Portuguese [no + ve], Yiddish, Yiddish [na + yn], [na + oi], [ti + sa]

10 (base 4, German [ze + hn], loglan [ne + ni], Spanish [di + ez], Swahili [ku + mi], Yiddish [ts + en])

8,918,432,389,728,768,545,100,000,000 (Goedel representation) = SSSS0 = g(2, 8, 2)g(2, 8, 3)g(2, 8, 5)g(2, 8, 7)g(2, 9, 11)

0.5 (modulos 7/2, 7/4, 7/8)

1.5 (bases 8/3, 7/2, modulo 5/2)

2.5 (bases 8/5, 7/4)

3.5 (bases 8/7, 7/6)

5.5 (bases 7/10, -1/2)

6.5 (bases 7/12, -1/4)

7.5 (bases 1/2, -1/6)

8.5 (base 7/16, -1/8)

9.5 (bases 7/18, -1/10)

phi/2 (modulos (15 + G(2, 1/2, 5)/2, (15 + G(2, 1/2, 5)/4, (15 + G(2, 1/2, 5)/8)

phi (modulo (7 + G(2, 1/2, 5)/2)

inversions (read upside-down):

2 teacups plus 2 teacups = 40 cups

6 teacups = 60 cups,

7 teacups = 70 cups,

8 teacups = 80 cups,

The Babylonians practiced the oldest known jootsy calculus. We still use a modified form of their system for counting hours, minutes and seconds, as in 12:59:59 + 1= 1. It is interesting in that unlike ours, had no radix point to indicate fractions. A numerical expression therefore could have multiple meanings depending upon context, 2 + 2 = 4 could actually mean 2/60 + 2/60 = 4/60 = 1/15, 2/3600 + 2/3600 = 4/3600 = 4/3600 = 1/900, etc.

Even more interesting is Babylonian division (like jootsing from hours to minutes to seconds and back without colons).

11/12 = 55 = 11(5), 9/10 = 54 = 9(6), 5/6 = 50 = 5(10), 4/5 = 48 = 4(12), 7/10 = 42 = 7(6), 2/3 = 40 = 2(20), 3/5 = 36 = 3(12), 1/2 = 30 = 10(1 + 2), 2/5 = 24 = 2(12), 1/3 = 20, 3/10 = 18 = 3(6), 1/4 = 15 = (1 + 4)(4 - 1), 1/5 = 12, 1/6 = 10 = 2(6 - 1), 1/15 = 4 = 5 - 1, 1/20 = 3 = 1 + 2 + 0

30*30 = 15, 30*30*30 = 450 = 30*15, 30*30*30*30 = 225 = 15*15, 30*30*30*30*30 = 6750 = 30*15*15, 30*30*30*30*30*30 = 3375 = 15*15*15, 30*30*30*30*30*30*30 = 101250 = 30*15*15*15

These are mathematical cryptograms in which letters are substituted for digits. TWO + TWO = FOUR could therefore mean many other things:

In base 10 we have seven possible answers:

734 765 836 846 867 928 938

+734 +765 +836 +846 +867 +928 +938

1468 1530 1672 1692 1734 1856 1876

which could just as validly be other words than "four"-- BOAS, BOAR, BODY, BOER, BOGS, BOGY, BOIL, BOLE, BOLA, BOLD, BONS, BONY, BONA, BOND, BONE, BONG, COAL, COAX, COBS, CODE, CODA, COGS, COIF, COIL, COIN, COKE, COLA, COLD, COMA, COMB, COME, CONE, CONY, CONK, etc.

Since this equation has only six symbols it could also have digits substituted for digits in any base six or above, only provided the three matching symbols are preserved, :

36 + 2(6) + 3 = 51 49 + 2(7) + 3 64 + 2(8) + 3

36 + 2(6) + 3 = 51 49 + 2(7) + 3 64 + 2(8) + 3 5(216) + 3(36) + 4(6) 5(343) + 3(49) + 4(7) 5(512) + 3(64) + 4(8)

= 1212 = 1890 = 2784

81 + 2(9) + 3 1331 + 2(121) + 3(11)

5(729) + 3(81) + 4(9) 5(14641) +3(121) + 4(11)

You can even add apples and oranges --

The most cryptic cryptarithms however are the ones that substitute digits for digits so that 2 + 2 = 4 could mean: 1 + 1 = 2, 2 + 2 = 4 or 4 + 4 = 8, and that's just in integral bases below 10.

The digital root is the end product of "casting out nines" or repeatedly summing the digits of a number. 2 + 2 which is 4 is also the digital root of:

13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 400, 1003, 1012, 1021, 1102, 1111, 1201, 1210, 1300, ...

Ethicalculus is a subdiscipline of jootsy calculus dealing with the hexadecimal pseudowords* formed by the alphanumeric characters -- 0 1 5 9 A B C D E F, where

A (base 16) = 10 (base 10),

B (base 16) = 11 (base 10),

C (base 16) = 12 (base 10),

D (base 16) = 13 (base 10),

E (base 16) = 14 (base 10),

F (base 16) = 15 (base 10)

100% 900D (base 16) = 36877 (base 10)

100% BAD (base 16) = 2989 (base 10)

and countless other alphanumeric calculations using the basic relationship for pseudowords between good and bad, better numbers (above 900d) and worse numbers (below BAD), many of which are four letters:

ABE = 92.004% BAD

A9E = 90.933% BAD

BAD cubed = 637B00E5D = 26,704,088,669 (base 10)

BAD squared = 8852E9 = 8,934,121 (base 10)

DAD = 1.511% 900D + 98.489% BAD

DO9 = 1.027% 900D + 98.973% BAD

E90 = 2.181% 900D + 97.819% BAD

F1B = 2.591% 900D + 97.409% BAD

F0E = 2.553% 900D + 97.447% BAD

900D + BAD = 8460 = 33,888 (base 10)

900D*BAD = 691E7C9 = 110,225,353 (base 10)

900D + 900D = 1201A = 73,754 (base 10)

900D cubed = 2A39C581D395 = 50,149,516,458,133 (base 10)

900D squared = 510EA0A9 = 1,359,913,129 (base 10)

half BAD = 5D6.8 = 1494.5 (base 10)

Square roots and cube roots are not the only roots any more than base ten is the only number system. The other polygonal roots just are not as well known. The triangular root, for example, is the length of the side of an equilateral triangle formed by successive gnomons (as in the additional series of linerally decreasing pile of "bricks"). Some of which are square roots of other numbers.

the triangular root of 10

the dodecagonal root of 64

the octodecagonal root of 100

Honest calculations

are, like the honest number four, honest in having the same letter count as the meaning of the expression.

one plus twelve = 1 + 12

seven plus seven = 7 + 7

two plus nine = 2 + 9

two plus twelve = 2 + 12

twenty-one divided by one = 21/1

twenty-three divided by one = 23/1

HONEST MULTIPLICATION

one times fifteen = 1*15

one times seventeen = 1*17

one multiplied by twenty-three = 1*23

HONEST SUBTRACTION

eighteen minus two = 18 - 2

sixteen minus one = 16 - 1

twenty minus five = 20 - 5

MYSTERIOUS ADDITION

From Louis Hoffman's Puzzles Old and New comes such challenging stuff as:

XI + X = XIX (11 + 10 = 19)

XX + I = XIX (20 + 1 = 19)

IV + VI = IX or XI (4 + 6 = 9 or 11)

(hint: read upside down)

MYSTERIOUS DIVISION

IX/2 = IV (9/2 = 4 or 1 or 10)

XII/2 = VII (12/2 = 7)

XL/2 = VI (40/2 = 6 or 10 or 50)

MYSTERIOUS SUBTRACTION

(or as Hoffman calls it subtraction extraordinary) has: "Take one from twenty and leave nineteen."

XIX - I = XX (19 - 1 = 20)

IX - X= I (9 - 10 = 1)

XIX - X = XI (19 - 10 = 11)

XIX - XI = X (19 - 11 = 10)

XIX - XX = I (19 - 20 = 1)

XL - L = X (40 - 50 = 10)

XLIX - XI = LX (49 - 11 = 50)

XLIX - XIX = L (49 - 19 = 50)

XLIX - XX = LI (49 - 20 = 51)

XLIX - LIX = X (49 - 59 = 10)

SIX - IX = S (6 - 9 = 1)

This can also be even more mysteriously done for addition in right-to-left base-eight Aramaean, rather than left-to-right base-ten Roman, numerals.

X - III = I (4 - 3 = 1)

X - II = II (4 - 2 = 2)

X - I = III (4 - 1 = 3)

IX - X = I (5 - 4 = 1)

IX - III = II (5 - 3 = 2)

IX - II = III (5 - 2 = 3)

XX - IX = III (8 - 5 = 3)

XX - III = IX (8 - 3 = 5)

SIX - IX = I (6 - 5 = 1)

means addition in binary without carry-over, as in the game, Nim.

1 + 1 = 0, 1 + 3 = 2, 1 + 5 = 4, 1 + 7 = 6, ...

2 + 2 = 0, 2 + 3 = 1, 2 + 6 = 4, 2 + 7 = 5, ...

3 + 1 = 2, 3 + 2 = 1, 3 + 3 = 0, 3 + 5 = 6, ...

4 + 4 = 0, 4 + 5 = 1, 4 + 6 = 2, 4 + 7 = 3, ...

NIM-MULTIPLICATION

means multiplication in binary without carry-over,

2*2 = 3, 2*3 = 1, 2*6 = 11, 2*7 = 9, 2*8 = 12, ...

3*2 = 1, 3*3 = 2, 3*6 = 13, 3*7 = 14, 3*8 = 4, ...

4*4 = 6, 4*5 = 2, 4*6 = 14, 4*7 = 10, 4*8 = 11, ...

5*4 = 2, 5*5 = 7, 5*6 = 8, 5*7 = 13, 5*8 = 3, ...

NUMBRAL ADDITION

means binary bitwise disjunction

1 + 1 = 1, 2 + 2 = 10 or 10 = 10 = 2,

NUMBRAL MULTIPLICATION

means binary bitwise shift-and-disjunction

2*2 = 100 or 10 = 110 = 6, 2*3 = 100 or 11 = 111 = 7, 3*2 = 110 or 10 = 6

Similar to mysterious addition is adding an odd number (of letters) to an odd number and getting an odd number.

Or adding odd to even to get even:

Subtracting odd from odd and getting odd:

or an even from an odd and getting an even:

or odd from even and getting an even:

even from even and getting an odd:

PALINDROMIC EQUATIONS

are equations that read the same backward and forward:

SEPTEMVIGINTARIES

These are based on Lee Sallows' system of transforming words and phrases into numbers via base 27, with A through Z evaluated at 1 through 26 and the space as 0, so that:

two = 20(729) + 23(27) + 15 = 15216 (base 10)

two + two = 30,432 (base 10) = antc

two plus two = 20(5559060566555523) + 23(205891132094649) + 15(7625597484987) + 16(10460353203) + 12(387420489) + 21(14348907) + 19(531441) + 20(729) + 23(27) + 15 = 116031263657698329 (base 10)

SEXATRIGINTARIES

In this base 36 system A = 10, B = 11, ..., Z = 35 so that we have:

TWO + TWO = 236G (corresponding to 2776 in base 10)

UNPROPER FRACTIONS

is where cancelation of digits actually works:

A wordnum is the sum of the letter values in a word, as in septemvirgintinaries or base 27 (without taking place value into account),

TWO + TWO = 2O + 2T + 2W = 58


	index André Joyce Fan Club \| Dominissimo \| Googology \| Hierogonometry \| Jootsy Calculus \| links \| Laws \| Merology \| neologisms \| non-Najunamarian \| Pataphysics \| Sequences \| transfinites \| trazillion \| zerology "In creative problem solving, it is frequently more important to look at a problem from different vantage points rather than run with the first solution that pops into your head." -- Eugene Raudsepp (Creative Growth Games) Jootsy Calculus in which 2 + 2 = 5, among other things translated by Michael Joseph Halm copyright 1978-2005 Hierogamous Enterprises Jootsy calculus is that branch of metamathematics in which calculations are not limited to just one alphanumeric system but allowed to JOOTS, "Jump Out Of The System" in which it began or had been previously "jootsed". The name is alternatively said to derive from that ancient practisioner, the Chinese philosopher and that protopataphysicist, Dzu-tse (also known by the Latinized Jucius and the Anglicized Jootsy). With jootsy calculus, for example, two plus two not only can equal five, but an infinite number of other correct answers, depending on the viewer's pov ["point of view"] even including infinities. One, in fact, can use any mathematical representation included in the set of Generalized Orthographic Denotations. "All things are possible with GOD." One can add two letters to two letters to get a number word in any number of languages, two letters to two letters to get a Roman numeral, two multiplicative factors to get a quotient, use a different base for the number system than the usual decimal, including "improper" representations using greater-than-base, half, imaginary or even irrational place holders, modulo arithmetic that notes only remainders. For example here are a few of the possible answers to what 2 + 2 = integers 0 (ze + ro, Nim-addition) 1 (French [on + ze], German [ei + ns], Yiddish [ey + ns], Italian [on + ce], Japanese [ic + hí , ka + ta], modulos 3, 3/2, Swahili [mo + ja], Russian [od + in, od + na]) 2 (French [de + ux], German [zw + ei], Italian [do + ce], Portuguese [do + is]) 3 (French and Spanish [tr + es], German [dr + ei], Swahili [ta + tu], Yiddish [dr + ay, fir]) 4 (bases 4+, Swedish [fy + ra], English [fo + ur], German [vi + er], Norwegian [fi + re], Roman [ij + ij]) 5 (English [fi + ve], French [ci + nq], Swahili [ta + no], Swedish [fy + ra], Yiddish [fi + nf]) 6 (Spanish [cuatro, se + is], Roman [ij + iv], French [quatre], German [se + is], Italian [si + ta], Norwegian [se + ks], Yiddish [ze + ks] 7 [French [se + pt], Italian [quattro], Portuguese [se + te], Spanish (sa + ba, Yiddish [zi + bn]) 8, [French [hu + it], na + ne, German [oc + ht], Italian [ot + to], Latin [quattuor], Portuguese [oi + to], Roman [vi + ij], Spanish [oc + ho)]) 9 [English [ni + ne], French [ne + uf], German [ne + un], (Japanese [ky + uu], Italian and Portuguese [no + ve], Yiddish, Yiddish [na + yn], [na + oi], [ti + sa] 10 (base 4, German [ze + hn], loglan [ne + ni], Spanish [di + ez], Swahili [ku + mi], Yiddish [ts + en]) 11 (base 3) 12 (base 2) 13 (base 1) 14 (base 0) 8,918,432,389,728,768,545,100,000,000 (Goedel representation) = SSSS0 = g(2, 8, 2)g(2, 8, 3)g(2, 8, 5)g(2, 8, 7)g(2, 9, 11) decimals 0.5 (modulos 7/2, 7/4, 7/8) 1.5 (bases 8/3, 7/2, modulo 5/2) 2.5 (bases 8/5, 7/4) 3.5 (bases 8/7, 7/6) 4.5 (base 7/8) 5.5 (bases 7/10, -1/2) 6.5 (bases 7/12, -1/4) 7.5 (bases 1/2, -1/6) 8.5 (base 7/16, -1/8) 9.5 (bases 7/18, -1/10) irrationals phi/2 (modulos (15 + G(2, 1/2, 5)/2, (15 + G(2, 1/2, 5)/4, (15 + G(2, 1/2, 5)/8) phi (modulo (7 + G(2, 1/2, 5)/2) imaginaries: 2i (base 2 - i/2) 3i (base (4 - i)/3) 4i (base 1 - i/4) . . . inversions (read upside-down): (9 - 8)(9 - 8) 9 - 8 + 9 - 8 (9 - 8)(98 - 88) 91 - 81 + 91 - 81) (9 - 8)(908 - 808) (91 - 81)(91 - 81)) . . . negatives: -10 (base -4) -11 (base -5) T-MATH 2 teacups plus 2 teacups = 40 cups 6 teacups = 60 cups, 7 teacups = 70 cups, 8 teacups = 80 cups, 9 teacups = 90 cups BABYLONIAN MATH The Babylonians practiced the oldest known jootsy calculus. We still use a modified form of their system for counting hours, minutes and seconds, as in 12:59:59 + 1= 1. It is interesting in that unlike ours, had no radix point to indicate fractions. A numerical expression therefore could have multiple meanings depending upon context, 2 + 2 = 4 could actually mean 2/60 + 2/60 = 4/60 = 1/15, 2/3600 + 2/3600 = 4/3600 = 4/3600 = 1/900, etc. Even more interesting is Babylonian division (like jootsing from hours to minutes to seconds and back without colons). 11/12 = 55 = 11(5), 9/10 = 54 = 9(6), 5/6 = 50 = 5(10), 4/5 = 48 = 4(12), 7/10 = 42 = 7(6), 2/3 = 40 = 2(20), 3/5 = 36 = 3(12), 1/2 = 30 = 10(1 + 2), 2/5 = 24 = 2(12), 1/3 = 20, 3/10 = 18 = 3(6), 1/4 = 15 = (1 + 4)(4 - 1), 1/5 = 12, 1/6 = 10 = 2(6 - 1), 1/15 = 4 = 5 - 1, 1/20 = 3 = 1 + 2 + 0 Or multiplication: 3030 = 15, 303030 = 450 = 3015, 30303030 = 225 = 1515, 3030303030 = 6750 = 301515, 303030303030 = 3375 = 151515, 30303030303030 = 101250 = 30151515 CRYPTARITHMETIC These are mathematical cryptograms in which letters are substituted for digits. TWO + TWO = FOUR could therefore mean many other things: In base 10 we have seven possible answers: 734 765 836 846 867 928 938 +734 +765 +836 +846 +867 +928 +938 1468 1530 1672 1692 1734 1856 1876 which could just as validly be other words than "four"-- BOAS, BOAR, BODY, BOER, BOGS, BOGY, BOIL, BOLE, BOLA, BOLD, BONS, BONY, BONA, BOND, BONE, BONG, COAL, COAX, COBS, CODE, CODA, COGS, COIF, COIL, COIN, COKE, COLA, COLD, COMA, COMB, COME, CONE, CONY, CONK, etc. Since this equation has only six symbols it could also have digits substituted for digits in any base six or above, only provided the three matching symbols are preserved, : 36 + 2(6) + 3 = 51 49 + 2(7) + 3 64 + 2(8) + 3 36 + 2(6) + 3 = 51 49 + 2(7) + 3 64 + 2(8) + 3 5(216) + 3(36) + 4(6) 5(343) + 3(49) + 4(7) 5(512) + 3(64) + 4(8) = 1212 = 1890 = 2784 81 + 2(9) + 3 1331 + 2(121) + 3(11) 81 + 2(9) + 3 1331 + 2(121) + 3(11) 5(729) + 3(81) + 4(9) 5(14641) +3(121) + 4(11) = 3924 = 73612 You can even add apples and oranges -- APPLES + ORANGES TUVWXYZ The most cryptic cryptarithms however are the ones that substitute digits for digits so that 2 + 2 = 4 could mean: 1 + 1 = 2, 2 + 2 = 4 or 4 + 4 = 8, and that's just in integral bases below 10. DIGITAL ROOTS The digital root is the end product of "casting out nines" or repeatedly summing the digits of a number. 2 + 2 which is 4 is also the digital root of: 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 400, 1003, 1012, 1021, 1102, 1111, 1201, 1210, 1300, ... ETHICALCULUS Ethicalculus is a subdiscipline of jootsy calculus dealing with the hexadecimal pseudowords* formed by the alphanumeric characters -- 0 1 5 9 A B C D E F, where A (base 16) = 10 (base 10), B (base 16) = 11 (base 10), C (base 16) = 12 (base 10), D (base 16) = 13 (base 10), E (base 16) = 14 (base 10), F (base 16) = 15 (base 10) 100% 900D (base 16) = 36877 (base 10) 100% BAD (base 16) = 2989 (base 10) and countless other alphanumeric calculations using the basic relationship for pseudowords between good and bad, better numbers (above 900d) and worse numbers (below BAD), many of which are four letters: ABE = 92.004% BAD A9E = 90.933% BAD BAD cubed = 637B00E5D = 26,704,088,669 (base 10) BAD squared = 8852E9 = 8,934,121 (base 10) DAD = 1.511% 900D + 98.489% BAD DO9 = 1.027% 900D + 98.973% BAD E90 = 2.181% 900D + 97.819% BAD F1B = 2.591% 900D + 97.409% BAD F0E = 2.553% 900D + 97.447% BAD 900D + BAD = 8460 = 33,888 (base 10) 900D*BAD = 691E7C9 = 110,225,353 (base 10) 900D + 900D = 1201A = 73,754 (base 10) 900D cubed = 2A39C581D395 = 50,149,516,458,133 (base 10) 900D squared = 510EA0A9 = 1,359,913,129 (base 10) half BAD = 5D6.8 = 1494.5 (base 10) GNOMONICS Square roots and cube roots are not the only roots any more than base ten is the only number system. The other polygonal roots just are not as well known. The triangular root, for example, is the length of the side of an equilateral triangle formed by successive gnomons (as in the additional series of linerally decreasing pile of "bricks"). Some of which are square roots of other numbers. [ ] [ ][ ] [ ][ ][ ] [ ][ ][ ][ ] 2 + 2 = the triangular root of 10 the dodecagonal root of 64 the octodecagonal root of 100 . . . Honest calculations are, like the honest number four, honest in having the same letter count as the meaning of the expression. HONEST ADDITION one plus twelve = 1 + 12 seven plus seven = 7 + 7 two plus nine = 2 + 9 two plus twelve = 2 + 12 HONEST DIVISION twenty-one divided by one = 21/1 twenty-three divided by one = 23/1 HONEST MULTIPLICATION one times fifteen = 115 one times seventeen = 117 one multiplied by twenty-three = 123 HONEST SUBTRACTION* eighteen minus two = 18 - 2 sixteen minus one = 16 - 1 twenty minus five = 20 - 5 MYSTERIOUS ADDITION From Louis Hoffman's Puzzles Old and New comes such challenging stuff as: XI + X = XIX (11 + 10 = 19) XX + I = XIX (20 + 1 = 19) IV + VI = IX or XI (4 + 6 = 9 or 11) (hint: read upside down) MYSTERIOUS DIVISION IX/2 = IV (9/2 = 4 or 1 or 10) XII/2 = VII (12/2 = 7) XL/2 = VI (40/2 = 6 or 10 or 50) 8/2 = 3 or 0 MYSTERIOUS SUBTRACTION (or as Hoffman calls it subtraction extraordinary) has: "Take one from twenty and leave nineteen." XIX - I = XX (19 - 1 = 20) Similarly: IX - X= I (9 - 10 = 1) XIX - X = XI (19 - 10 = 11) XIX - XI = X (19 - 11 = 10) XIX - XX = I (19 - 20 = 1) XL - L = X (40 - 50 = 10) XLIX - XI = LX (49 - 11 = 50) XLIX - XIX = L (49 - 19 = 50) XLIX - XX = LI (49 - 20 = 51) XLIX - LIX = X (49 - 59 = 10) SIX - IX = S (6 - 9 = 1) This can also be even more mysteriously done for addition in right-to-left base-eight Aramaean, rather than left-to-right base-ten Roman, numerals. X - III = I (4 - 3 = 1) X - II = II (4 - 2 = 2) X - I = III (4 - 1 = 3) IX - X = I (5 - 4 = 1) IX - III = II (5 - 3 = 2) IX - II = III (5 - 2 = 3) XX - IX = III (8 - 5 = 3) XX - III = IX (8 - 3 = 5) SIX - IX = I (6 - 5 = 1) NIM-ADDITION means addition in binary without carry-over, as in the game, Nim. 1 + 1 = 0, 1 + 3 = 2, 1 + 5 = 4, 1 + 7 = 6, ... 2 + 2 = 0, 2 + 3 = 1, 2 + 6 = 4, 2 + 7 = 5, ... 3 + 1 = 2, 3 + 2 = 1, 3 + 3 = 0, 3 + 5 = 6, ... 4 + 4 = 0, 4 + 5 = 1, 4 + 6 = 2, 4 + 7 = 3, ... NIM-MULTIPLICATION means multiplication in binary without carry-over, 22 = 3, 23 = 1, 26 = 11, 27 = 9, 28 = 12, ... 32 = 1, 33 = 2, 36 = 13, 37 = 14, 38 = 4, ... 44 = 6, 45 = 2, 46 = 14, 47 = 10, 48 = 11, ... 54 = 2, 55 = 7, 56 = 8, 57 = 13, 58 = 3, ... NUMBRAL ADDITION means binary bitwise disjunction 1 + 1 = 1, 2 + 2 = 10 or 10 = 10 = 2, NUMBRAL MULTIPLICATION means binary bitwise shift-and-disjunction 22 = 100 or 10 = 110 = 6, 23 = 100 or 11 = 111 = 7, 32 = 110 or 10 = 6 ODD ADDITION* Similar to mysterious addition is adding an odd number (of letters) to an odd number and getting an odd number. 7 + -ty-one = 71 Or adding odd to even to get even: sty + even = 70 ODD SUBTRACTION Subtracting odd from odd and getting odd: 71 - (-ty-one) = 7 or an even from an odd and getting an even: 61 - (--y-one) = 60 or odd from even and getting an even: 400 - hundred = 4 even from even and getting an odd: 70 - ty = 7 PALINDROMIC EQUATIONS are equations that read the same backward and forward: 1242 = 2421 1263 = 3621 1284 = 4821 1362 = 2631 1393 = 3931 1482 = 2841 2364 = 4632 2396 = 6932 2463 = 3642 2484 = 4842 2693 = 3962 3486 = 6843 3684 = 4863 4696 = 6964 112422 = 224211 112633 = 336211 112844 = 448211 SEPTEMVIGINTARIES These are based on Lee Sallows' system of transforming words and phrases into numbers via base 27, with A through Z evaluated at 1 through 26 and the space as 0, so that: two = 20(729) + 23(27) + 15 = 15216 (base 10) two + two = 30,432 (base 10) = antc two plus two = 20(5559060566555523) + 23(205891132094649) + 15(7625597484987) + 16(10460353203) + 12(387420489) + 21(14348907) + 19(531441) + 20(729) + 23(27) + 15 = 116031263657698329 (base 10) SEXATRIGINTARIES In this base 36 system A = 10, B = 11, ..., Z = 35 so that we have: TWO + TWO = 236G (corresponding to 2776 in base 10) UNPROPER FRACTIONS is where cancelation of digits actually works: 16/64 = 16/64 = 1/4 19/95 = 19/95 = 1/5 26/65 = 26/65 = 2/5 49/98 = 49/98 = 4/8 64/16 = 64/16 = 4/1 65/26 = 65/26 = 5/2 WORDNUMS A wordnum is the sum of the letter values in a word, as in septemvirgintinaries or base 27 (without taking place value into account), TWO + TWO = 2O + 2T + 2W = 58