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AS066/KH0079 - Group Sum Bands
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**Drawings:**

**Group Sum Bands:**

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**Individual Group Sum Bands:**-

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# | Group Sum Bands Schema | Group Position 1 Based Index |
Split Index 1 Based Index |
Left Sum 1 Based Index |
Right Sum 1 Based Index |
---|---|---|---|---|---|

0 | 036 | 3 | 25 = W@[37, 1]:13 + W@[37, 2]:12 | 25 = W@[37, 3]:13 + W@[37, 4]:12 | |

1 | 052 | 3 | 26 = W@[53, 1]:13 + W@[53, 2]:13 | 26 = W@[53, 3]:14 + W@[53, 4]:12 | |

2 | 055 | 5 | 100 = MB@[56, 1]:25 + MB@[56, 2]:25 + MB@[56, 3]:31 + MB@[56, 4]:19 | 100 = MB@[56, 5]:25 + MB@[56, 6]:25 + MB@[56, 7]:25 + MB@[56, 8]:25 | |

3 | 057 | 5 | 25 = KB@[58, 1]:6 + KB@[58, 2]:6 + KB@[58, 3]:7 + KB@[58, 4]:6 | 25 = W@[58, 5]:6 + W@[58, 6]:6 + W@[58, 7]:7 + W@[58, 8]:6 |

**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 â‰ P5 + P6 + P7 + P8, it is interesting to note that also:

\[ P_{1}+P_{2}+P_{3}+P_{4} = \frac{top\_value}{2}-6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;DOWN \]\[ P_{5}+P_{6}+P_{7}+P_{8} = \frac{top\_value}{2}+6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;UP \]