AS066/KH0079 - Pendant Pendant Sums

Drawings:

Overview	Detail View
All Khipus' Ascher Sum Fieldmarks	Right-Handed XRay
All Pendant Pendant Sum Relationships	Left-Handed XRay
This Khipu's Fieldmark Catalog	Right-Handed Sum Detail
This Khipu's Drawing	Left-Handed Sum Detail

Right Handed Sums: # Sums = 25, Max # Summands = 9, (Min, Mean, Max) Sum Values = (12, 32, 160)
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Left Handed Sums: # Sums = 15, Max # Summands = 6, (Min, Mean, Max) Sum Values = (12, 25, 72)
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Right Handed Sum Detail: - Click on column name to sort

#	Sum Cord	Sum Cord Value	# Summands	Summands
1	p3_{3, 1} : 160_W	160	9	p14: 16_W + p15: 15_W + p16: 24_W + p17: 9_W + p18: 72_W + p19: 9_W + p20: 9_W + p21: 0_KB + p22: 6_W
2	p4_{3, 2} : 101_W	101	6	p25: 10_KB + p26: 30_KB + p27: 8_MB + p28: 50_KB + p29: 0_KB + p30: 3_MB
3	p12_{7, 2} : 21_W	21	2	p63: 6_MB + p64: 15_MB
4	p15_{9, 2} : 15_W	15	3	p20: 9_W + p21: 0_KB + p22: 6_W
5	p16_{9, 3} : 24_W	24	4	p19: 9_W + p20: 9_W + p21: 0_KB + p22: 6_W
6	p18_{10, 2} : 72_W	72	5	p23: 8_W + p24: 16_W + p25: 10_KB + p26: 30_KB + p27: 8_MB
7	p24_{13, 1} : 16_W	16	4	p109: 4_KB + p110: 4_MB + p111: 4_MB + p112: 4_MB
8	p36_{20, 4} : 30_MB	30	2	p42: 13_MB + p43: 17_MB
9	p39_{22, 2} : 40_MB	40	5	p60: 6_MB + p61: 6_MB + p62: 7_MB + p63: 6_MB + p64: 15_MB
10	p42_{25, 1} : 13_MB	13	2	p51: 6_MB + p52: 7_MB
11	p45_{25, 4} : 17_MB	17	2	p135: 12_W + p136: 5_MB
12	p81_{37, 4} : 12_W	12	2	p92: 6_W + p93: 6_W
13	p90_{40, 4} : 13_W	13	2	p91: 7_W + p92: 6_W
14	p101_{44, 2} : 28_MB	28	7	p109: 4_KB + p110: 4_MB + p111: 4_MB + p112: 4_MB + p113: 4_MB + p114: 4_MB + p115: 4_MB
15	p102_{44, 3} : 29_MB	29	2	p108: 25_MB + p109: 4_KB
16	p103_{44, 4} : 15_MB	15	3	p136: 5_MB + p137: 5_MB + p138: 5_MB
17	p108_{46, 4} : 25_MB	25	3	p130: 12_MB + p131: 0_PK + p132: 13_W
18	p121_{49, 4} : 18_W	18	3	p157: 6_KB + p158: 6_W + p159: 6_W
19	p124_{49, 7} : 32_W	32	5	p135: 12_W + p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB
20	p125_{49, 8} : 31_W	31	3	p134: 14_W + p135: 12_W + p136: 5_MB
21	p133_{53, 2} : 13_W	13	2	p155: 6_KB + p156: 7_KB
22	p135_{53, 4} : 12_W	12	2	p154: 6_KB + p155: 6_KB
23	p147_{56, 3} : 31_MB	31	5	p154: 6_KB + p155: 6_KB + p156: 7_KB + p157: 6_KB + p158: 6_W
24	p148_{56, 4} : 19_MB	19	3	p154: 6_KB + p155: 6_KB + p156: 7_KB
25	p152_{56, 8} : 25_MB	25	4	p154: 6_KB + p155: 6_KB + p156: 7_KB + p157: 6_KB

Left Handed Sum Detail: - Click on column name to sort

#	Sum Cord	Sum Cord Value	# Summands	Summands
1	p18_{10, 2} : 72_W	72	3	p9: 24_KB + p10: 24_KB + p11: 24_W
2	p33_{20, 1} : 40_MB	40	2	p25: 10_KB + p26: 30_KB
3	p64_{32, 1} : 15_MB	15	3	p20: 9_W + p21: 0_KB + p22: 6_W
4	p69_{34, 1} : 13_MB	13	2	p62: 7_MB + p63: 6_MB
5	p71_{34, 3} : 12_MB	12	2	p60: 6_MB + p61: 6_MB
6	p100_{44, 1} : 28_MB	28	3	p62: 7_MB + p63: 6_MB + p64: 15_MB
7	p105_{46, 1} : 25_MB	25	4	p96: 7_MB + p97: 6_MB + p98: 6_MB + p99: 6_MB
8	p118_{49, 1} : 31_W	31	6	p58: 6_MB + p59: 0_KB + p60: 6_MB + p61: 6_MB + p62: 7_MB + p63: 6_MB
9	p121_{49, 4} : 18_W	18	3	p97: 6_MB + p98: 6_MB + p99: 6_MB
10	p129_{51, 3} : 14_MB	14	2	p22: 6_W + p23: 8_W
11	p130_{51, 4} : 12_MB	12	3	p114: 4_MB + p115: 4_MB + p116: 4_MB
12	p132_{53, 1} : 13_W	13	2	p96: 7_MB + p97: 6_MB
13	p145_{56, 1} : 25_MB	25	5	p139: 5_MB + p140: 5_MB + p141: 5_MB + p142: 5_MB + p143: 5_MB
14	p147_{56, 3} : 31_MB	31	3	p134: 14_W + p135: 12_W + p136: 5_MB
15	p148_{56, 4} : 19_MB	19	3	p96: 7_MB + p97: 6_MB + p98: 6_MB

Khipu Notes:

Ascher Databook Notes:

This khipu is associated with AS059-KH0080. See discussion under AS059.
From the discoloration of the main cord, and by extrapolation based on the spacing of the complete groups, we hypothesize that this khipu contained 24 groups of 8 pendants.
Each of the groups had a top cord. Of the original 80 pendants of the first 10 groups, only 35 remain and so the first complete group starts with pendant 36. Of the last 14 groups, no pendants are missing.
Two loose broken cords were associated with this khipu: one, colored DB, had a value of 50; the other, colored AB, had a value of 14.
Both common forms of top cord attachment are present on this khipu. In groups 18, 19, and 21-24, the top cord unites the pendants. In the remaining groups, the top cord is attached to the main cord in the center of the group.
The first pendant of the 18th group (P92) is a different color than the rest of the group reinforcing the idea that this is a place where a change occurs.
All pendant cords within a group are the same color with the exception of group 18 noted above and groups 16, 20, and 23 in which the first 4 pendants are one color and the last 4 another color.
In 2 of these groups the top cord is the same color as the first 4 pendants in the group. The third group has the top cord missing so its color is unknown.
Only 6 groups have the top cord the same color as all the pendants in the group. Five of these groups are the only groups for which all pendants in the group have the same value.
The color of the top cords for groups 8-11 (BB, W, DB, W) are repeated for the next 4 groups (12-15).
With the exception of group 19, all groups that have all pendant values and top cord value present show the following relationship:
\[ \sum\limits_{j=1}^4 P_{ij} = \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=13,15,16,17,18,21,22,23,24) \]
Where one of the three parts of the relationship is unknown due to breakage, the other two still appear:
\[ \sum\limits_{j=1}^4 P_{ij} = \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=12,14,17) \]

\[ \sum\limits_{j=1}^4 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=9) \]

\[ \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=2,10,11) \]

(Note: P_ij is the value of the j^th pendant in the i^th group.)
As noted in observation 6, five groups have all equal pendant values. The sum value on the top cord must, therefore, be a multiple of 8.
In some cases where the top value is not a multiple of 8, the pendant values can be viewed as top value/ 8 rounded to the nearest integers.
This is seen in groups 11,12,16,23 where the sums are 50. The pendant values are 6x6 + 2x7.
Similarly in group 9, the pendant values can be viewed as rounded to the nearest multiple of 10. That is 150 = 3x40 + 1x30.
Groups 17 and 19 both have the same relationship among the first 4 pendants:
\[ P_{1} = P_{2} = \frac{x}{4} + 3 \]

\[ P_{3} = \frac{x}{4} + 4 \]

\[ P_{4} = \frac{x}{4} - 10 \]
And as a result:
\[ P_{1}+P_{2} = \frac{x}{2} + 6,\;\;\;\;\;\;\;P_{3}+P_{4} = \frac{x}{2} - 6\]
For group 17, where X=100 and for group 19, where X=112.
Since group 19 is the only group for which P1 + P2 + P3 + P4 ≠ P5 + P6 + P7 + P8, it is interesting to note that also:
\[ P_{1}+P_{2}+P_{3}+P_{4} = \frac{top\_value}{2}-6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;DOWN \]

\[ P_{5}+P_{6}+P_{7}+P_{8} = \frac{top\_value}{2}+6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;UP \]