Slot Machine Arithmetic:

Casting Out Cherries

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Given:

An ORANGE is a number.

An ORANGE is NOT the successor of a number.

However...

The successor of an ORANGE is a LEMON.

The successor of a LEMON is a BAR.

The successor of a BAR is a BELL.

The successor of a BELL is CHERRIES.

The successor of CHERRIES is a LEMON_ORANGE.

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Counting in in Slot Machine Arithmetic (SMA):
ORANGE
LEMON
BAR
BELL
CHERRIES
LEMON_ORANGE
LEMON_LEMON
LEMON_BAR
LEMON_BELL
LEMON_CHERRIES
BAR_ORANGE
BAR_LEMON
BAR_BAR
BAR_BELL
BAR_CHERRIES
BELL_ORANGE
BELL_LEMON
BELL_BAR
BELL_BELL
BELL_CHERRIES
CHERRIES_ORANGE
CHERRIES_LEMON
CHERRIES_BAR
CHERRIES_BELL
CHERRIES_CHERRIES

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Addition in SMA:

            ORANGE   + ORANGE   = ORANGE
            ORANGE   + LEMON    = LEMON
            ORANGE   + BAR      = BAR
            ORANGE   + BELL     = BELL
            ORANGE   + CHERRIES = CHERRIES

            LEMON    + LEMON    = BAR
            LEMON    + BAR      = BELL
            LEMON    + BELL     = CHERRIES
        LEMON    + CHERRIES = LEMON_ORANGE

            BAR      + BAR      = CHERRIES
        BAR      + BELL     = LEMON_ORANGE
        BAR      + CHERRIES = LEMON_LEMON

        BELL     + BELL     = LEMON_LEMON
        BELL     + CHERRIES = LEMON_BAR

        CHERRIES + CHERRIES = LEMON_BELL

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To add larger numbers, step through the numbers one column at a time and look up each pair of digits in our addition table.

Example: what is the sum of ORANGE_LEMON_CHERRIES and CHERRIES_BAR_BELL?

Step 1: line up the two numbers vertically

Step 2: sum the digits in the right column.

CHERRIES + BELL = LEMON_BAR

Put the BAR in the right column and carry the LEMON.

Step 3: sum the digits in the middle column (two at a time).

LEMON + LEMON = BAR

BAR + BAR = CHERRIES

Put the CHERRIES in the middle column (there is no carry).

Step 4: sum the digits in the left column.

ORANGE + CHERRIES = CHERRIES

Put the CHERRIES in the LEFT column (there is no carry).

This completes the sum.

Therefore, ORANGE_LEMON_CHERRIES + CHERRIES_BAR_BELL = CHERRIES_CHERRIES_BAR

In standard arithmetic, we could test the validity of our answer by a process known as "Casting Out Nines". The algorithm applies to ALL number systems. We just need to know what it is and to identify what has to be "cast out".

"Casting Out Nines" is used for Base 10 arithmetic because "9" is the last item in the symbol set. The value of the last symbol in a number system is the Base-1 and is always the number that preceeds "10". Keep in mind that "10" is NOT "ten". It is ONE_ZERO, which is 1*Base^1 + 0*Base^0. In any base, 10^n mod Base-1 = 1. This is why we can "cast out" the digits. Each digit of a number represents the remainder when the modulo of the Base-1 is taken.

In Base 10, for 1234:

1000 mod 9 = 1; 200 mod 9 = 2; 30 mod 9 = 3; 4 mod 9 = 4

For other number systems, the item to "cast out" is always Base-1, which, of course, is the last symbol in the symbol set.

For Base 8 (octal)

symbol set {0 1 2 3 4 5 6 7}

cast out "7"

For Base 16 (hexadecimal)

synbol set {0 1 2 3 4 5 6 7 8 9 A B C D E F}

cast out "F"

Therefore, for SMA, we must "Cast Out CHERRIES".

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The Casting Out Nines algorithm:

Verify that 123 + 456 = 579


[1]	from the list of all the digits of all the operands,

[1a]	cast out all 9s and all sets of integers adding to nine

	from 123456 cast out: no 9s, the 4+5 pair and the 6+3 pair (or the 6+2+1 triple)
	
	leaving 12

[1b]	sum the remaing digits

	1 + 2 = 3

[1c]	if sum > 8, return to [1a]

[1d]	the number that's left is the REMAINDER (3)



[2]	cast out the 9s from the answer

[2a]	from 579 cast out: one 9

	leaving 57

[2b]	5 + 7 = 12

[2a]	from 12 cast out: nothing

	leaving 12

[2b]	1 + 2 = 3

[2d]	REMAINDER (3)



[3]	if the REMAINDER of the operands matches the REMAINDER of the answer (3=3),
	the "casting out nines" test has succeeded in verifying your answer.



Note:	"Casting Out Nines" only proves that your answer is CONSISTENT, i.e.,
	has the correct digits. Those digits would need to be in the correct
	order for your answer to be ACCURATE. Other tests, such as "Casting
	Out Elevens", could refine the confidence in the answer, but that is
	beyond the scope of this document.

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Applying the algorithm to SMA:

[1] cast out CHERRIES from the operands

[1a] from

     cast out: two and the , pair; leaving

[1b] sum the reaining digits:

[1d] REMAINDER ()

[2] cast out CHERRIES from the answer

[2a] from

     cast out: two ; leaving

[2d] REMAINDER ()

[3] REMAINDER () of operands = REMAINDER() of answer

Answer is CONSISTENT!

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Let's try another...

CHERRIES + BELL = LEMON_BAR

LEMON + LEMON = BAR

BAR + ORANGE = BAR

BELL + CHERRIES = LEMON_BAR

carry the LEMON + BAR + CHERRIES = LEMON_BAR

carry the LEMON + LEMON + BELL = LEMON_ORANGE

Now, from the operands...

...cast out...

three CHERRIES

three LEMON, BELL pairs, ,

one BAR, BAR pair

...leaving...

 REMAINDER of operands

From the answer...

...cast out...

two BAR, BAR pairs,

...leaving...

...sum the remaining digits...

 REMAINDER of answer

Aha!

The REMAINDER() of the operands doesn't match
the REMAINDER() of the answer, so the answer must be WRONG!

Can you see the mistake? I forgot to "carry the LEMON" when I added the first column of digits.

The answer should have been

...which, after casting out...

two BAR, BAR pairs,

one LEMON, BELL pair

...leaves...

 REMAINDER of answer

The REMAINDER() of the CORRECT answer matches
the REMAINDER() of the operands!

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