# 12th Dynasty KAHUN FRAGMENTS introduction and analysis:

Presented by Bruce C. Friedman

Page images created: January 2002

Before I begin analysis of the Kahun IV, 3 (Arithmetic Progression) you should closely review the example below to become familiar with my terminology.
Lets begin with a linear progression of numbers: { 3, 7, 11, 15 }

The sum of the four given numbers is 3+7+11+15 = 36
The Quantity is 36. Q=36

There are obviously the four given divisions of the Quantity.
The number of divisions is Portions.
The Portions is 4. P=4

Between the four portions are three spaces or gaps.
The number of gaps equals the number of portions minus one.
G=(P-1).
G=3

Each gap is the same. This is the definition of an arithmetic progression.
Each gap is the absolute value of the distance from two adjacent portions.
(15-11) and (11-7) and (7-3) each equal 4.
The typical gap (TG) is 4.
TG=4

The spread of the progression is the distance value [absolute value] of
the highest portion less the lowest portion.
15-3=12
Note: The Spread is equal to twice the MV
[or G(TG)]
S=12

The largest Portion is 15.
Hi is 15.

The smallest portion is 3.
Lo is 3.

The Quantity divided by the number of Portions is the Mean Average.
MA is 36/4.
MA is 9.
Note: MA = (Hi+Lo)/2.

The absolute value of the distance of the Hi or Lo portions from the MA is the Mean Variant.
MV= I Hi - MA I and also = I MA - Lo I.
MV= (15 - 9) and also = (9 - 3).
MV is 6.
Note MV = [G(TG)]/2 = [3(4)]/2 = 6

The Mean Variant squared is MV^2.
MV^2 = 6^2
MV^2 is 36

The product of the Hi and Lo portions is HILO.
HILO is 15 X 3.
HILO is 45.

The sum of the Hi and Lo portions is Hi+Lo.
Hi+Lo is 15 + 3.
Hi+Lo is 18
Note (Hi+Lo)/2 = MA.

Now you are ready to proceed!

What is noteworthy about the Kahun IV, 3 columns 11 and 12 data?

Verify this for yourself!
I find that unlike the example above, the Kahun IV, 3 progression satisfies
the following expression:
HILO + MV^2 = Q
Or, in familiar terms, we note that:
The product of the largest portion and the smallest portion, when added to the square of the distance either has from the average portion, is equal to the original quantity. This relationship in the Kahun IV, 3 arithmetic progression (A.P.) could be satisfied in infinite ways.
Try changing the highest portion and typical gap [without changing the number or portions or total quantity] to verify this.

In the above given [non-historical] example, when we test for the value of HILO+MV^2 we find:
[(15) X (3)] + [(6)^2] = 81. This does not equal the (36) quantity.
If we again test for a different A.P.
Same number of portions  and quantity  distributed as follows:
{ 7.5, 8.5, 9.5, 10.5 } we find that HILO+MV^2 is again 81, and will always be.
Test it now to see if you have grasped the above terms.

Note: HILO+MV^2=Q can only be satisfied when Portions^2=Quantity!
Why do you think this Q and P were chosen?
See the geometric analysis of the Kahun Fragment.

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