A brief biography of the Aschers is provided in the Ascher Relations Overview page. Thanks to Manuel Medrano's suggestion that their theories be investigated in depth, I have extracted the useful equations from the Ascher databooks, and mathematically typeset their typewritten notes in LaTex, for each Ascher khipu that that I was able to extract from the Harvard KDB. This enables viewing the relationships visually, and it provides another view into how inter-related are cord values in a khipu. For example, in the databook notes, the more lines that have = signs in them, the more the cord values depend on each other. This provides another useful metric besides Benford match, or sum clusters, for categorizing a khipu's typology.
A first pass at the compiled relationships is below. For more information, and to see the equations in their appropriate context, click on the link to the khipu on the left column.
| Khipu Name: | Equations: | ||||||||||||||||||||||||||||||||||||||||||||
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| AS010 |
P1+ P1s1= 51
\[ \sum\limits_{i=2}^{8} P_{i} = 58\; \]
\[ \sum\limits_{i=9}^{16} P_{i} = 51\; \]
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| AS020 |
Represent the value on the pendant in group i at position j by
\[V_{ij}\;\;where\;i=1,2;\;\;j=1,2,...,5 \]
Comparison between groups:
\[V_{1j}>V_{2j}\;for\;j=1,3,4,5 \]
Comparison within groups:
\[V_{i1}>V_{i2}\;\;and\;\;V_{i1}>V_{i3}\;>\;V_{i4}>V_{i5}\;for\;i=1,2 \]
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| AS026B |
\[N, N, {N \over 2}, N, N, N, N, N, {N \over 2}, {N \over 2} \]
Where there are 9 rather than 10 pendant positions, one of the 5 consecutive positions with value N is non-existent. The values involved are:
\[N=24; ({N \over 2}=12); N=9; ({N \over 2}=4); N=5; ({N \over 2}=4); N=8; ({N \over 2}=4); N=4; ({N \over 2}=2); N=8; ({N \over 2}=4); N=4; ({N \over 2}=2); N=2; ({N \over 2}=1)\]
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| AS029 |
\[P_{i}-P_{i,s1}=P_{j}\;\;\;\;for\;(i,j)=(5,4),\;(7,8),\;(9,10),\;(10,4)\]
\[P_{i,s1}-P_{i}=P_{j}\;\;\;\;for\;(i,j)=(1,2),\;(2,1),\;(4,1)\]
\[P_{i}=P_{i+5}+P_{i+10}\;\;for\;i=(1,2,3,4) \]
In keeping with its being their sum, Pi has subsidiaries of the same colors and values as Pi+5 and Pi+10 combined. Only one position shows a summation instead: a value of 55+7 appears while the subsidiaries it is adding are 20 and 42. (Of the 32 subsidiaries on the 12 pendants, 3 are not accounted for by this addition.) |
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| AS038 |
\[P_{2i}=P_{4i}+P_{5i}+P_{6i}\;for\;i=(1,2,3...,12) \]
\[P_{2,13}=P_{5,13}+P_{6,13} \]
\[P_{2,i}=P_{4,i-1}+P_{5i}+P_{6i}\;for\;i=(14,15,...,18) \]
\[P_{1i}=P_{2i}+P_{3i}\;for\;i=(1,2,3...,18) \]
\[P_{2,i°sub_j}=P_{3,i°sub_j}+P_{4,i°sub_j}+P_{5,i°sub_j}+P_{6,i°sub_j}\;for\;i=(1,2,3...,18)\;\;\;j=(B,W,B,W,GG,RL,BD,BS) \]
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| AS041 |
The values in Group 1 are the sums of the values in the subsequent groups. Where pendants of the same color are in similar positions in different groups, their values are related. However, the sum seems to be more related to pendant color than position. While other sum statements might be as valid, one which accounts for most values follows. (Note: Pij is the value of the jth pendant in the ith th group. Pijsk is the value of the kth subsidiary on the jth pendant in the ith group.)
\[P_{11}+P_{11s1}+P_{11s2}=P_{21}+P_{31}+P_{41}+P_{51}+P_{61}+P_{71}\;for\;all\;(LD) \]
\[P_{12}=P_{23}+P_{43}+P_{53}\;for\;all\;(MB) \]
\[P_{12s1}=P_{43s1}\;for\;all\;(W) \]
\[P_{13s1}=P_{52}+P_{22s1}+P_{24s1}+P_{44s1}\;for\;all\;(RL) \]
\[^{(RL)}{P_{13}=P_{22}+P_{32}+P_{92}}\;+\;^{(W)}{P_{63}}\;+\;^{(LD:W)}{P_{24}+P_{54}}\;for\;all\;(LD:W)\]
\[P_{14s1}=P_{32s1}\;for\;all\;(LD:W)\]
\[^{(LD:W)}{P_{14}}=\;^{(LD:W)}{P_{22}}\;+\;^{(LD-W)}{P_{73}}\;+\;^{(RL:W)}{P_{64}+P_{72}+P_{72s1}+P_{81}+P_{83}}\;for\;all\;(LD:W)\]
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| AS048 | position 2 has the maximum value. With the exception of the last group, P2> P1+P3+P4. | ||||||||||||||||||||||||||||||||||||||||||||
| AS066 |
\[ \sum\limits_{j=1}^4 P_{ij} = \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=13,15,16,17,18,21,22,23,24) \]
\[ \sum\limits_{j=1}^4 P_{ij} = \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=12,14,17) \]
\[ \sum\limits_{j=1}^4 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=9) \]
\[ \sum\limits_{j=5}^8 P_{ij} = \frac{top\_value}{2}\;\;\;for\;(i=2,10,11) \]
(Note: ij is the value of the jth pendant in the ith group.)
\[ P_{1} = P_{2} = \frac{x}{4} + 3 \]
\[ P_{3} = \frac{x}{4} + 4 \]
\[ P_{4} = \frac{x}{4} - 10 \]
\[ P_{1}+P_{2} = \frac{x}{2} + 6,\;\;\;\;\;\;\;P_{3}+P_{4} = \frac{x}{2} - 6\]
\[ P_{1}+P_{2}+P_{3}+P_{4} = \frac{top\_value}{2}-6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;DOWN \]
\[ P_{5}+P_{6}+P_{7}+P_{8} = \frac{top\_value}{2}+6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;UP \]
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| AS068 |
There are some additional relationships between sums of 5 consecutive groups in section C of the khipu. If Pijk> represents the value of the pendant in the ith part (i=l,2 ), the jth group of section C (j=l,2,...,10), and the kth position in the group (k=l,2,...,7), the relationships can be represented as follows:
\[ \sum\limits_{j=1}^5 P_{1j5} = \sum\limits_{j=6}^{10} P_{2j4} \]
\[ \sum\limits_{j=5i-4}^{5i} (P_{1j1}-P_{2j1}) = P_{i,10,1}\;\;\;for\;i=(1,2) \]
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| AS079 |
\[P_{i1}=P_{i3}\;\;for\;i\;=\;(10,11,12,13,14)\]
\[P_{i1}=P_{i2}-1\;\;for\;i\;=\;(10,11,12,14,15)\]
\[P_{i1}=P_{i2}-1\;=P_{i3}\;for\;i\;=\;(10,11,12,14)\]
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| AS080 |
The values in groups 1 and 3 are related in reverse order. Let P1i and P3i (i=1,2,...,6) represent values in groups 1 and 3 respectively and s1i and s3i
\[P_{1i}+P_{3,7-i}=11\;\;for\;i\;=\;(2,3,4,5)\]
\[S_{1i}+S_{3,7-i}=7\;\;for\;i\;=\;(3,4,5)\]
\[S_{1i}=+S_{3,7-i}\;\;for\;i\;=\;(1,6)\]
\[V_{1i}\;\;for\;i\;=\;(1,2,...,11)\]
\[V_{3i}\;\;for\;i\;=\;(1,2,...,14)\]
\[V_{1i}=V_{3i}\;\;for\;i\;=\;(1,2,3)\]
\[V_{1i}=V_{3,i+3}\;\;for\;i\;=\;(4,5,6,8,9,10,11)\]
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| AS082 |
\[X_{1}+X_{16}=12\]
\[X_{2}+X_{15}=18\]
\[X_{3}+X_{14}=24\]
\[X_{4}+X_{13}=30\]
\[X_{i}+X_{17-i}=6(i+1)\;\;for\;i\;=\;(2,3,4)\]
Xi and X17-i are both divisible by YXi+l and X17-(i+l) are relatively prime Xi+2 and X17-(i+2) are equal, both are divisible by Y and Xi+2/Y = 3 Xi+3 and X17-(i+3) are both divisible by Y-2 |
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| AS092 |
\[ P_{1i} > P_{4i} > P_{3i} > P_{2i}\;for\;\;i\;=\;(2,3,4,...,8) \]
\[ P_{4i} + P_{5i} = P_{6i} + P_{7i}\;for\;\;i\;=\;(1,4) \]
\[ P_{2i} + P_{3i} = P_{6i} + P_{7i}\;for\;\;i\;=\;(4,6) \]
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| AS115 |
\[ \sum\limits_{i=1}^3 P_{i4} = \sum\limits_{i=4}^7 P_{i5} = 1000 \]
\[ \sum\limits_{i=1}^7 P_{i4} = \sum\limits_{i=1}^7 P_{i5} \]
\[ \sum\limits_{j=4}^7 P_{2j} = \sum\limits_{j=4}^7 P_{3j} \]
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| AS128 |
Pendants 1-3 of groups 3-5 have the same relationships to each other as do pendants 5-7 of these same groups. In each case, the 9 pendants form the pattern:
Where a=3 when Pi=P1 and a=4 when Pi=P5 |
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| AS129 |
Pi7 > Pi2 > Pi4 > Pi3 > Pi5 > Pi6 for i=(1, 2) With the exception of the 6th position, the values of group 2 are greater than those in corresponding positions in group 1. P2j > P1j for j=(1, 2...,5,7) The last value in group 1 is the sum of 3 other values in the group: P17 = P11 + P13 + P16 Two pairs of values in group 1 have the same sum. In both pairs, the values are 3 positions apart. P11 + P14 = P13 + P16 With a discrepancy of 1, the sum is the same for a pair of values in corresponding positions in both groups. P13 + P16 ≈ P23 + P26 |
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| AS164 |
\[ P_{11}s1 = \sum\limits_{i=1}^{4} P_{i1} ~~~~~~ P_{14}s1 = \sum\limits_{j=1}^{6} P_{3j} ~~~~~~ P_{14}s2 = \sum\limits_{j=1}^{6} P_{2j}s1 \]
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| AS168 |
\[ \sum\limits_{i=1}^{9} P_{2i-1} = \sum\limits_{i=1}^{9} P_{2i} \]
where Pi is the value of the ith pendant.
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| AS192 |
\[ P_{1} = \sum\limits_{i=3}^{10} P_{i} \]
\[ P_{1s1} = \sum\limits_{i=3}^{6} P_{i} \]
\[ P_{1s2} = \sum\limits_{i=3}^{5} P_{i} \]
\[ P_{1}s_{3} = \sum\limits_{i=3}^{5} (P_{i} + P_{12-i}) \] P1s1= 49 = 72 P1s3= 64 = 82 P3= 16 = 42 P10= 36 = 62 Yi i=(1,...,5) (i. e., Y1= P2; Y2= P3+ P4+ P5; Y3= P6; Y4= P7+ P8+ P9; Y5=P10) Y1+ Y3+ Y5= 92 Y2+ Y4= 82 Y4= 52 Y5= 62 Y2+ Y3= 72 Y2+ Y4= 82 Y2+ Y4+ Y5= 102
\[ Y_{2}+Y_{3} = \sum\limits_{i=3}^{6} P_{i} = {7}^{2} \]
\[ \sum\limits_{i=3}^{6} (P_{i})^{2} = {25}^{2} \]
P1= Y2 + Y3 + Y4 + Y5 P1 s1 = Y2 + Y3 P1 s2 = Y2 P1 s3 = Y2 + Y4 An alternate separation into subgroups of 3, 1, 1, 1, 3 pendants such that: Y1=P2+P3+P4 Y2=P5 Y3=P6 Y4=P7 Y5=P8+P9+P10 gives: Y1+Y3+Y5=112 Finally, the sum cord P1 and its subsidiaries can be viewed in terms of squares. P1 =52 + 62 + 72 P1 s3 =82 P1 - P1 s1 =52 + 62 P1 s3-P1 s2 = 52 or P1 - P1 s1 + P1 s2 = 62 + 82 =102 P1 s3 - P1 s2 + P1 s1 = 52 + 72 P1 - P1 s1+P1 s2 - P1 s3 = 62 -(P1 s3 - P1 s2 + P1 s1) + P1 = 62 |
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| AS193 | P1+P2= P5+P6, and P3+P4= P7+P8. | ||||||||||||||||||||||||||||||||||||||||||||
| AS197 | P1+P2= P3+P4+P5+P6+P7+P8. | ||||||||||||||||||||||||||||||||||||||||||||
| AS198 |
P5= P6 P1= P2= P3= P4= P8= 4
\[ \sum\limits_{i=1}^{9} P_{i} = 32 \]
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| AS199 |
\[ P_{8j}=\sum\limits_{i=1}^{4} P_{ij}\;\;\;for\;j=1 \]
\[ P_{11,j+1} = \sum\limits_{i=1}^{4} P_{ij}\;\;\;for\;j=4 \]
\[ P_{9,j+1}=\sum\limits_{i=1}^{4} P_{ij}\;\;\;for\;j=(2,4) \]
\[ P_{5,j+1}=\sum\limits_{i=1}^{4} P_{ij}\;\;\;for\;j=3 \]
\[ P_{11,j+1}=\sum\limits_{i=1}^{4} P_{ij}\;\;\;for\;j=5 \]
\[ P_{3,j-2}=\sum\limits_{i=5}^{8} P_{ij}\;\;\;for\;j=5 \]
\[ \sum\limits_{i=5}^{8} P_{ij} = \sum\limits_{i=5}^{8} P_{i,j+3}\;\;\;for\;j=3 \]
P11,j= P5j+ P8j= P6j+ P9j for j=4 P5j= P2j+ P6j for j=(1,2,..., 6) |
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| AS201 |
P3> P1> P2> P4 |
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| AS206 | P5> P4> P6> P1> P3≥ P2. | ||||||||||||||||||||||||||||||||||||||||||||
| AS207A |
\[ P_{127} = \sum\limits_{i=1}^{6} P_{12i} \]
\[ P_{132} = \sum\limits_{i=3}^{8} P_{13i} \]
P122= P212 P123= P216 P135= P214 P13,i+5= P21i for i=1,2,3 Since we hypothesize that the P21i are sums of the next 9 groups in Part II, P132 would be a sum of sums.
\[ P_{21j} = \sum\limits_{i=2}^{10} P_{2ij} \]
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| AS208 |
Pi1= 0 17 ≤ Pi6= max(j) Pij≤ 72 Pi2 and Pi5≤ 32< a Pi3 and Pi4≤ 13 Pi6> Pi2> Pi5> Pi3> Pi4> Pi1 P2j> P1j P3j< P2j P4j> P3j P5j< P4j P6j> P5j P7j< P6j f> P3j+ P4j= P7j for j=(3, 6, 10) P6j= P7j for j=(13, 14, 15) |
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| AS209 |
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| AS210 |
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| AS211 |
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| AS212 |
P1ij For i=(1,2); j=(1,...,6) (with j≠(1,2) for i=1) with subsidiaries P1ijs for i=1; j=(1,2) P2ij for i=(1,...,4); j=(1,...6) (j≠3 for i=1,2 and j≠5 for i=3) with subsidiaries P2ijs for all i and j=(1,2) P4ij i=(1,...,4); j=(2,...,6) fori=1,3; and j=(1,...,6) for i=(2,4) with subsidiaries P4ijk where k=1 for j=(1,2); k=(0,1,2) for j=3; and k=(0,1) for j=(4,5,6) For i=(2,4) P4i1= 2 P4i2+1 (or[P412/2]=P4i1); For j=(2,...,6) P43j= 2 P44j |
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| AS213 |
Calling the pendants Pij where i=(1,2) are the subgroups and j= i9) the positions in the subgroup P1j is even for j=(1,...,9) P1j ≥ P2j for j=(1,3,5,7,9) Pij < P2j for j=(2,4,6,8) |
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| AS214 | 2*P1j= P2,7-j for j=(1,2,3) | ||||||||||||||||||||||||||||||||||||||||||||
| AS215 |
P3ij where i=(1,...,4) is the group and j=(1,...,4) is pendant position in group. 11 ≥ P3i2> P3i4≥ P3i3= P3i1= 1 for i=(1,...,4)
\[ P_{11} = \sum\limits_{j=1}^{4} P_{31j} = \sum\limits_{j=1}^{4} P_{33j} \]
Calling the six pendant values in Part II P21j where j=(1,...,6):
\[ P_{211} = 4 = \sum\limits_{j=1}^{4} P_{31j}\;\;\;\;(or \sum\limits_{j=1}^{4} P_{33j}) \]
\[ P_{212} = 14 = P_{11} \]
\[ P_{213} = 7 = \sum\limits_{j=1}^{4} P_{34j} \]
\[ P_{214} = 37 = \sum\limits_{j=1}^{4} P_{32j} \]
\[ P_{215} = 10 = P_{322}\;subsidiary \]
\[ P_{216} = 4 = \sum\limits_{j=1}^{4} P_{33j}\;\;\;\;(or \sum\limits_{j=1}^{4} P_{31j}) \]
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| UR1034 |
\[P_{i1}=P_{i2}\;\;for\;i\;=\;(3,4)\]
\[P_{i2}=P_{i3}\;\;for\;i\;=\;(3,5,6)\]
\[P_{i4}=P_{i5}\;\;for\;i\;=\;(2,3,4,5,6)\]
For groups 8 and 9, all pendant values are multiples of 3 with most of them being 30. Specifically,
\[P_{8i}=P_{9i}=30\;\;for\;i\;=\;(1,2,3,5)\]
\[P_{i}=P_{i+1},\;P_{i+2}=P_{i+3}\;\;for\;i=(1,\;4)\;in\;group\;11\;and\;i=1\;in\;group\;13 \]
\[P_{i}+1=P_{i+3},\;P_{i+3}+1=P_{i+6}\;\;for\;i=(1,\;2) \]
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| UR1096 |
\[ P_{13}\;≥\;P_{i1}+P_{i2}+P_{i4}+P_{i5}+P_{i6} \;\;\;for\;(i=2,3,4,...,10) \]
\[ P_{13}\;=\;2(P_{i1}+P_{i2}+P_{i4}+P_{i5}+P_{i6})\;\;\;for\;(i=2,3,4) \]
\[ P_{7i}\;=\;\sum\limits_{j=4}^6 P_{ji}\;\;\;for\;(i=1,2,3,4) \]
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| UR1097 |
\[ P_{2,2i} = 5\;for\;\;i\;=\;(1,...,9) \]
\[ P_{3,2i} = 5\;for\;\;i\;=\;(1,...,10) \]
\[ P_{2,2i-1} = 1\;for\;\;i\;=\;(4,..,9) \]
\[ P_{3,2i-1} = 1\;for\;\;i\;=\;(4,...,10) \]
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| UR1098 |
\[ \sum\limits_{i=1}^2 P_{i2} = 48\&1 \]
\[ \sum\limits_{i=3}^6 P_{i2} = 46 \]
\[ \sum\limits_{i=1}^6 i= 94\&1\;\;\;P_{1}=94\&1 \]
Pi2 is the 2'nd pendant in the ith pair. \[ \sum\limits_{i=1}^3 P_{2i} = 48\&1 \]
\[ \sum\limits_{i=4}^6 P_{2i} = 46 \]
\[ \sum\limits_{i=1}^6 P_{2i} = 94\&1\;\;\;P_{1}=94\&1 \]
P2i is the ith pendant in the 2'nd group.The value 18&2 also appears on AS0174 and is the sum of the subsidiaries of the first pendant in the 1th pair. P1s1s1=18&2&1 and is color W;
\[ \sum\limits_{i=1}^6 P_{i1}s1s1 = 18\&2 \]
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| UR1100 |
\[ \sum\limits_{i=1}^6 P_{i2} = \sum\limits_{j=1}^{10} P_{2j} \]
\[ \sum\limits_{i=1}^6 P_{i4} = \sum\limits_{j=1}^9 P_{4j} \]
\[ \sum\limits_{i=1}^6 P_{i3} = \sum\limits_{i=1}^6 P_{i7} \]
\[ \sum\limits_{i=1}^6 P_{i5} = \sum\limits_{i=1}^6 P_{i6} \]
\[ \sum\limits_{i=1}^6 P_{i8} = \sum\limits_{i=1}^6 P_{i10} \]
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| UR1103 |
\[ P_{2i}\;=\;P_{3i}+P_{3i}s1+P_{4i}\;\;\;for\;i=(7,9) \]
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| UR1104 |
\[ P_{14}\;=\;P_{16}+5300\;=\;P_{33}+22000 \]
\[ P_{16}\;=\;P_{33}+16700 \]
\[ P_{11}\;=\;P_{42}+1200\;=\;P_{44}+1900 \]
\[ P_{21}\;=\;P_{35}+600 \]
\[ P_{31}\;=\;P_{41}+400 \]
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| UR1114 |
P11, P46, P56, P76, P87, P91, P98-P9,10, P10,6, P10,8- P10,10, P13,1- P13,3, and P13,7- P13,9 are all 0. The same non-zero values occur in both groups in the following positions: P12-P16, P31, P34, P71, P12,6, P12,8, and P14,2 Thus, 90 of 160 positions have the same pendant values. |
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| UR1116 |
P2i-1≥ P2i+1 for i=1, 2, 3, 4 P2i≥ P2i+2 for i=1, 2, 3, 4, 5 Pi> Pi+1 for i=1, 3, 5, 7, 11 |
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| UR1117 |
P1= P3+ P4 P4= P5+ P6 P6= P9+ P10+ P11 P7= P8+ P9+ P10+ P11
\[ P_{2} = \sum\limits_{i=4}^{11} P_{i}^2 \]
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| UR1120 |
\[ \sum\limits_{i=2}^4 P_{ij} = P_{1j}\;\;\;\;\;for\;\;j=(1,2,...,8) \]
p2j/P1j= 0.342 max. deviation 1.2% p3j/P1j= 0.425 max. deviation 0.7% p4j/P1j= 0.235 max. deviation 0.9% |
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| UR1126 |
\[ \sum\limits_{i=1}^6 P_{i1}=\sum\limits_{i=1}^6 P_{i2}=\sum\limits_{i=1}^6 P_{i3}= 250 \]
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| UR1131 |
\[ \sum\limits_{j=1}^{7} P_{1j3} = \sum\limits_{j=1}^{7} P_{1j4} \]
\[ \sum\limits_{j=1}^{7} P_{1j2} = \sum\limits_{j=1}^{7} P_{2j4} \]
\[ \sum\limits_{j=1}^{7} P_{1j6} = \sum\limits_{j=1}^{7} P_{2j2} \]
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| UR1135 | Pi7> Pi6> Pi8> Pi5 for i=(1,2,3) | ||||||||||||||||||||||||||||||||||||||||||||
| UR1136 |
\[ \sum\limits_{i=1}^{10} (P_{i4}+P_{i5}) = 433 \]
\[ \sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j} \]
\[ \sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j} = \frac{1}{2} \sum\limits_{j=1}^{9} P_{7j}\]
\[ \sum\limits_{j=1}^{9} (P_{4j} + P_{5j}) = \sum\limits_{j=1}^{9} P_{7j}\]
\[ \sum\limits_{j=2}^{9} P_{6j} = 142 \]
\[ \sum\limits_{i=6}^{10} P_{i2} = \sum\limits_{i=6}^{10} P_{i3} \]
\[ \sum\limits_{j=1}^{9} (P_{2j}+P_{7j}) = \sum\limits_{j=1}^{9} (P_{3j}+P_{8j}) \]
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| UR1138 |
Pi1= Pi2 Pi3> Pi4> Pi5 Pi36=Pi7
\[ \frac{P_{i7}}{P_{i1}} \approx 2.4\;\;\;for\;i=(2,3,4,5)\]
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| UR1140 |
\[ \sum\limits_{i=1}^{6} P_{i5} = \sum\limits_{i=1}^{6} P_{i6} \]
\[ \sum\limits_{i=7}^{10} P_{i1} = \sum\limits_{i=7}^{10} P_{i2} \]
\[ \sum\limits_{j=1}^{8} P_{1j} = \sum\limits_{j=1}^{8} P_{7j} \]
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| UR1141 |
\[ \sum\limits_{i=1}^{10} P_{i1}=\sum\limits_{i=1}^{10} P_{i2} \]
Pi1= Pi2 for i=(1,3,5,6,...,10) Pi1= Pi2= 22 for i=(3,5,6,...,10) |
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| UR1143 |
\[ P_{1j} = \sum\limits_{i=2}^{5} P_{ij}\;\;\;for\;j=(1,2,3,4,5) \]
Pi3> Pi1> Pi5> Pi2> Pi4 for i=(1,2,3,4,5) P1j> P4j> P3j≥ P5j> P2j for j=(1,2,3,4,5) |
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| UR1145 |
P3i+ P3,i+1= P1i for i=(1,4) P3i+ P1i= 2 P3,i-1 for i=(4,7) |
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| UR1148 |
\[ \sum\limits_{i=1}^{4} P_{2ij} = P_{11j}\;\;\;for\;j=(1,2) \]
\[ \sum\limits_{i=1}^{4} P_{3ij} = P_{12j}\;\;\;for\;j=(1,2) \]
where Pkij is the value of the pendant in the jth position of the ith group of the kth part.P2i > P2i-1 i=(1,...,10) and P2i > P2i+1 i=(1,...,9) where Pi is the value of the ith pendant on the khipu. |
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| UR1149 |
\[ P_{21j} = \sum\limits_{i=1}^{9} P_{3ij} \;\;\;for\;j=(1,...,5) \]
\[ P_{11j} = \sum\limits_{i=1}^{7} P_{2ij}\;\;\;\;for\;all\;j=(1,...,5)\]
\[ P_{12j} = \sum\limits_{i=8}^{14} P_{2ij}\;\;\;\;for\;all\;j=(1,...,5) \]
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| UR1151 |
P21= 13
\[ \sum\limits_{i=3}^{6} P_{2i}\;=\;13 \]
P22= 13
\[ \sum\limits_{i=3}^{6} P_{2i}\;subsidiaries\;=\;13 \]
P3,11= 26
\[ \sum\limits_{i=1}^{5} P_{3i}\; = \;26 \]
\[ \sum\limits_{i=1}^{5} P_{3i}\;\;subsidiaries\; = \;P_{3,11}\;subsidiary \]
P3,12= 26
\[ \sum\limits_{i=6}^{10} P_{3i}\; = \;26 \]
\[ \sum\limits_{i=1}^{5} P_{3i}\;\;subsidiaries\; = \;P_{3,12}\;subsidiary \]
\[ \sum\limits_{i=1}^{10} P_{3i}\;\;subsidiaries\; = \;P_{3,11}\;subsidiary + P_{3,12}\;subsidiary = 26 \]
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| UR1152 |
\[ P_{310} = \sum\limits_{j=1}^{3} P_{3ij}\;\;\;for\;i=(1,4) \]
P2,i,2j-1= P3,j,i+ P3,j+3,i for j=(1,2,3) and i=(1,2,3)
\[ P_{5i0} = \sum\limits_{j=1}^{4} P_{5ij}\;\;\;for\;i=(1,4) \]
P4,i,j= P5,j,i+ P5,j+3,i for j=(1,2) and i=(1,2,3,4) |
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| UR1163 |
Pi= Pi+1 i=(5,10,11) Pi= Pi+2 i=(1,10) Pi= Pi+2+ Pi+3 i=(2,3,5,7,8) but not i=(1,4,6 9) Pi* P13-i= 36 i=(1,3,4,5) but not i=(2,6)
\[ \frac{P_{i}}{P_{i+1}}=\frac{P_{12-i}}{P_{13-i}}\;\;\;for\;i=(3,4) \]
Pi= Pi+1 i=(3,7,9) Pi= Pi+2 i=(10) Pi= Pi+2+ Pi-1 i=(3,7) Pi* P13-i= 0 i=(1,3,4,5) but not i=(2,6)
\[ P_{1} = \sum\limits_{i=2}^{6}P_{2i-1}=\sum\limits_{i=2}^{6}P_{2i} \]
Pi= Pi+2+ Pi+c i=(3,7) for Group 1, c=3 while for Group 2, c=1 Pi* P13-i= K i=(1,3,4,5) for Group 1, K=36 while for Group 2, K=0. P1> P2> P3≥ P4> P5 P7≥ P8> P9≥ P10 |
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| UR1175 |
Pij= Pi+1,j and the latter as Pij= Pi+2,j
\[ P_{2ij}= \sum\limits_{k=0}^{6} P_{3,3k+i,j}\;\;\;for\;j=(1,2,...5),\;\; i=(1,2,3) \]
This represents 15 sums and 105 values being summed. Of the 15 values in part 2, 8 are exactly these sums (or off by 1 in 1 digit); 5 are exact sums of only some of the 7 pendants: Example: P211 = P341 + P3,10,1 + P3,13,1 + P3,16,1 + P3,19,1 |
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| UR1180 |
P2,2= P2,4+ P2,12= P2,7+ P2,9+ P2,10+ P2,11 P2,10= P2,9+ P2,11 P1,1= P2,2+ P2,7+ P2,9+ P2,10+ P2,12 P1,5= P2,2+ P2,3+ P2,7= P2,3+ P2,4+ P2,7+ P2,12 |
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