AS066/KH0079 - Colored Pendant Sums

Drawings:

Overview	Detail View
All Khipus' Ascher Sum Fieldmarks	Right-Handed XRay
All Colored 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 = 43, Max # Summands = 10, (Min, Mean, Max) Sum Values = (12, 47, 300)
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Left Handed Sums: # Sums = 34, Max # Summands = 10, (Min, Mean, Max) Sum Values = (12, 46, 300)
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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	8	p14: 16_W + p15: 15_W + p16: 24_W + p17: 9_W + p18: 72_W + p19: 9_W + p20: 9_W + p22: 6_W
2	p4_{3, 2} : 101_W	101	7	p89: 13_W + p90: 13_W + p91: 7_W + p92: 6_W + p93: 6_W + p94: 6_W + p95: 50_W
3	p11_{7, 1} : 24_W	24	3	p19: 9_W + p20: 9_W + p22: 6_W
4	p12_{7, 2} : 21_W	21	4	p160: 7_W + p161: 6_W + p163: 4_W + p164: 4_W
5	p13_{8, 1} : 170_KB	170	3	p28: 50_KB + p41: 110_KB + p77: 10_KB
6	p14_{9, 1} : 16_W	16	4	p163: 4_W + p164: 4_W + p165: 4_W + p166: 4_W
7	p16_{9, 3} : 24_W	24	3	p19: 9_W + p20: 9_W + p22: 6_W
8	p18_{10, 2} : 72_W	72	9	p133: 13_W + p134: 14_W + p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W + p161: 6_W + p163: 4_W + p164: 4_W
9	p24_{13, 1} : 16_W	16	4	p163: 4_W + p164: 4_W + p165: 4_W + p166: 4_W
10	p33_{20, 1} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
11	p34_{20, 2} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
12	p35_{20, 3} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
13	p36_{20, 4} : 30_MB	30	3	p45: 17_MB + p51: 6_MB + p52: 7_MB
14	p37_{21, 1} : 300_W	300	10	p95: 50_W + p118: 31_W + p119: 31_W + p120: 32_W + p121: 18_W + p122: 31_W + p123: 31_W + p124: 32_W + p125: 31_W + p132: 13_W
15	p38_{22, 1} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
16	p39_{22, 2} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
17	p50_{27, 1} : 50_W	50	4	p73: 13_W + p74: 13_W + p75: 13_W + p76: 11_W
18	p64_{32, 1} : 15_MB	15	3	p136: 5_MB + p137: 5_MB + p138: 5_MB
19	p65_{32, 2} : 15_MB	15	3	p136: 5_MB + p137: 5_MB + p138: 5_MB
20	p66_{32, 3} : 15_MB	15	3	p136: 5_MB + p137: 5_MB + p138: 5_MB
21	p68_{33, 1} : 102_W	102	8	p78: 13_W + p79: 12_W + p80: 13_W + p81: 12_W + p82: 13_W + p83: 13_W + p84: 13_W + p85: 13_W
22	p71_{34, 3} : 12_MB	12	3	p110: 4_MB + p111: 4_MB + p112: 4_MB
23	p79_{37, 2} : 12_W	12	3	p163: 4_W + p164: 4_W + p165: 4_W
24	p81_{37, 4} : 12_W	12	3	p163: 4_W + p164: 4_W + p165: 4_W
25	p100_{44, 1} : 28_MB	28	7	p110: 4_MB + p111: 4_MB + p112: 4_MB + p113: 4_MB + p114: 4_MB + p115: 4_MB + p116: 4_MB
26	p101_{44, 2} : 28_MB	28	7	p110: 4_MB + p111: 4_MB + p112: 4_MB + p113: 4_MB + p114: 4_MB + p115: 4_MB + p116: 4_MB
27	p103_{44, 4} : 15_MB	15	3	p136: 5_MB + p137: 5_MB + p138: 5_MB
28	p104_{45, 1} : 200_MB	200	8	p145: 25_MB + p146: 25_MB + p147: 31_MB + p148: 19_MB + p149: 25_MB + p150: 25_MB + p151: 25_MB + p152: 25_MB
29	p105_{46, 1} : 25_MB	25	5	p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB + p140: 5_MB
30	p106_{46, 2} : 25_MB	25	5	p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB + p140: 5_MB
31	p107_{46, 3} : 25_MB	25	5	p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB + p140: 5_MB
32	p108_{46, 4} : 25_MB	25	5	p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB + p140: 5_MB
33	p117_{48, 1} : 32_MB	32	5	p130: 12_MB + p136: 5_MB + p137: 5_MB + p138: 5_MB + p139: 5_MB
34	p118_{49, 1} : 31_W	31	4	p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W
35	p119_{49, 2} : 31_W	31	4	p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W
36	p120_{49, 3} : 32_W	32	3	p134: 14_W + p135: 12_W + p158: 6_W
37	p121_{49, 4} : 18_W	18	4	p161: 6_W + p163: 4_W + p164: 4_W + p165: 4_W
38	p122_{49, 5} : 31_W	31	4	p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W
39	p123_{49, 6} : 31_W	31	4	p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W
40	p124_{49, 7} : 32_W	32	3	p134: 14_W + p135: 12_W + p158: 6_W
41	p125_{49, 8} : 31_W	31	4	p135: 12_W + p158: 6_W + p159: 6_W + p160: 7_W
42	p134_{53, 3} : 14_W	14	3	p161: 6_W + p163: 4_W + p164: 4_W
43	p135_{53, 4} : 12_W	12	3	p163: 4_W + p164: 4_W + p165: 4_W

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

#	Sum Cord	Sum Cord Value	# Summands	Summands
1	p37_{21, 1} : 300_W	300	10	p4: 101_W + p11: 24_W + p12: 21_W + p14: 16_W + p15: 15_W + p16: 24_W + p17: 9_W + p18: 72_W + p19: 9_W + p20: 9_W
2	p42_{25, 1} : 13_MB	13	3	p27: 8_MB + p30: 3_MB + p31: 2_MB
3	p64_{32, 1} : 15_MB	15	4	p27: 8_MB + p30: 3_MB + p31: 2_MB + p32: 2_MB
4	p65_{32, 2} : 15_MB	15	4	p27: 8_MB + p30: 3_MB + p31: 2_MB + p32: 2_MB
5	p66_{32, 3} : 15_MB	15	4	p27: 8_MB + p30: 3_MB + p31: 2_MB + p32: 2_MB
6	p69_{34, 1} : 13_MB	13	3	p27: 8_MB + p30: 3_MB + p31: 2_MB
7	p70_{34, 2} : 13_MB	13	3	p27: 8_MB + p30: 3_MB + p31: 2_MB
8	p72_{34, 4} : 13_MB	13	3	p27: 8_MB + p30: 3_MB + p31: 2_MB
9	p86_{39, 1} : 104_W	104	5	p18: 72_W + p19: 9_W + p20: 9_W + p22: 6_W + p23: 8_W
10	p95_{42, 1} : 50_W	50	4	p73: 13_W + p74: 13_W + p75: 13_W + p76: 11_W
11	p100_{44, 1} : 28_MB	28	3	p62: 7_MB + p63: 6_MB + p64: 15_MB
12	p101_{44, 2} : 28_MB	28	3	p62: 7_MB + p63: 6_MB + p64: 15_MB
13	p103_{44, 4} : 15_MB	15	4	p27: 8_MB + p30: 3_MB + p31: 2_MB + p32: 2_MB
14	p105_{46, 1} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
15	p106_{46, 2} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
16	p107_{46, 3} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
17	p108_{46, 4} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
18	p117_{48, 1} : 32_MB	32	3	p67: 6_MB + p69: 13_MB + p70: 13_MB
19	p120_{49, 3} : 32_W	32	4	p19: 9_W + p20: 9_W + p22: 6_W + p23: 8_W
20	p121_{49, 4} : 18_W	18	3	p92: 6_W + p93: 6_W + p94: 6_W
21	p124_{49, 7} : 32_W	32	4	p19: 9_W + p20: 9_W + p22: 6_W + p23: 8_W
22	p126_{50, 1} : 237_MB	237	9	p30: 3_MB + p31: 2_MB + p32: 2_MB + p33: 40_MB + p34: 40_MB + p35: 40_MB + p36: 30_MB + p38: 40_MB + p39: 40_MB
23	p130_{51, 4} : 12_MB	12	3	p110: 4_MB + p111: 4_MB + p112: 4_MB
24	p144_{55, 1} : 40_MB	40	3	p98: 6_MB + p99: 6_MB + p100: 28_MB
25	p145_{56, 1} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
26	p146_{56, 2} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
27	p147_{56, 3} : 31_MB	31	3	p44: 8_MB + p45: 17_MB + p51: 6_MB
28	p148_{56, 4} : 19_MB	19	3	p51: 6_MB + p52: 7_MB + p53: 6_MB
29	p149_{56, 5} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
30	p150_{56, 6} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
31	p151_{56, 7} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
32	p152_{56, 8} : 25_MB	25	4	p51: 6_MB + p52: 7_MB + p53: 6_MB + p54: 6_MB
33	p153_{57, 1} : 200_KB	200	4	p25: 10_KB + p26: 30_KB + p28: 50_KB + p41: 110_KB
34	p171_{61, 1} : 32_W	32	3	p134: 14_W + p135: 12_W + p158: 6_W

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:
\[ P_{1}+P_{2}+P_{3}+P_{4} = \frac{top\_value}{2}-6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;DOWN \]

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