# The Ascher Databook Equations

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:
AS010 P1+ P1s1= 51

$\sum\limits_{i=2}^{8} P_{i} = 58\;$

$\sum\limits_{i=9}^{16} P_{i} = 51\;$
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:
$V_{i1}>V_{i2}\;\;and\;\;V_{i1}>V_{i3}\;>\;V_{i4}>V_{i5}\;for\;i=1,2$
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:
$N=24; ({N \over 2}=12); N=9; ({N \over 2}=4); N=5; ({N \over 2}=4); N=8; ({N \over 2}=4); N=4; ({N \over 2}=2); N=8; ({N \over 2}=4); N=4; ({N \over 2}=2); N=2; ({N \over 2}=1)$
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.)
AS038
$P_{2i}=P_{4i}+P_{5i}+P_{6i}\;for\;i=(1,2,3...,12)$

$P_{2,13}=P_{5,13}+P_{6,13}$

$P_{2,i}=P_{4,i-1}+P_{5i}+P_{6i}\;for\;i=(14,15,...,18)$

$P_{1i}=P_{2i}+P_{3i}\;for\;i=(1,2,3...,18)$

$P_{2,i°sub_j}=P_{3,i°sub_j}+P_{4,i°sub_j}+P_{5,i°sub_j}+P_{6,i°sub_j}\;for\;i=(1,2,3...,18)\;\;\;j=(B,W,B,W,GG,RL,BD,BS)$
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.)

$P_{11}+P_{11s1}+P_{11s2}=P_{21}+P_{31}+P_{41}+P_{51}+P_{61}+P_{71}\;for\;all\;(LD)$

$P_{12}=P_{23}+P_{43}+P_{53}\;for\;all\;(MB)$

$P_{12s1}=P_{43s1}\;for\;all\;(W)$

$P_{13s1}=P_{52}+P_{22s1}+P_{24s1}+P_{44s1}\;for\;all\;(RL)$

$^{(RL)}{P_{13}=P_{22}+P_{32}+P_{92}}\;+\;^{(W)}{P_{63}}\;+\;^{(LD:W)}{P_{24}+P_{54}}\;for\;all\;(LD:W)$

$P_{14s1}=P_{32s1}\;for\;all\;(LD:W)$

$^{(LD:W)}{P_{14}}=\;^{(LD:W)}{P_{22}}\;+\;^{(LD-W)}{P_{73}}\;+\;^{(RL:W)}{P_{64}+P_{72}+P_{72s1}+P_{81}+P_{83}}\;for\;all\;(LD:W)$

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.)

$P_{1} = P_{2} = \frac{x}{4} + 3$

$P_{3} = \frac{x}{4} + 4$

$P_{4} = \frac{x}{4} - 10$

$P_{1}+P_{2} = \frac{x}{2} + 6,\;\;\;\;\;\;\;P_{3}+P_{4} = \frac{x}{2} - 6$

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

$P_{5}+P_{6}+P_{7}+P_{8} = \frac{top\_value}{2}+6\;\;\;where\;\frac{top\_value}{2}\;is\;rounded\;UP$
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:
$\sum\limits_{j=1}^5 P_{1j5} = \sum\limits_{j=6}^{10} P_{2j4}$

$\sum\limits_{j=5i-4}^{5i} (P_{1j1}-P_{2j1}) = P_{i,10,1}\;\;\;for\;i=(1,2)$
AS079
$P_{i1}=P_{i3}\;\;for\;i\;=\;(10,11,12,13,14)$

$P_{i1}=P_{i2}-1\;\;for\;i\;=\;(10,11,12,14,15)$

$P_{i1}=P_{i2}-1\;=P_{i3}\;for\;i\;=\;(10,11,12,14)$
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
$P_{1i}+P_{3,7-i}=11\;\;for\;i\;=\;(2,3,4,5)$

$S_{1i}+S_{3,7-i}=7\;\;for\;i\;=\;(3,4,5)$

$S_{1i}=+S_{3,7-i}\;\;for\;i\;=\;(1,6)$

$V_{1i}\;\;for\;i\;=\;(1,2,...,11)$

$V_{3i}\;\;for\;i\;=\;(1,2,...,14)$

$V_{1i}=V_{3i}\;\;for\;i\;=\;(1,2,3)$

$V_{1i}=V_{3,i+3}\;\;for\;i\;=\;(4,5,6,8,9,10,11)$
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 Y
Xi+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
AS092
$P_{1i} > P_{4i} > P_{3i} > P_{2i}\;for\;\;i\;=\;(2,3,4,...,8)$

$P_{4i} + P_{5i} = P_{6i} + P_{7i}\;for\;\;i\;=\;(1,4)$

$P_{2i} + P_{3i} = P_{6i} + P_{7i}\;for\;\;i\;=\;(4,6)$
AS115
$\sum\limits_{i=1}^3 P_{i4} = \sum\limits_{i=4}^7 P_{i5} = 1000$

$\sum\limits_{i=1}^7 P_{i4} = \sum\limits_{i=1}^7 P_{i5}$

$\sum\limits_{j=4}^7 P_{2j} = \sum\limits_{j=4}^7 P_{3j}$
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:

Group Pi Pi+1 Pi+2
Group 3 a+1 a+1 a+1
Group 4 a a a
Group 4 a a a

Where a=3 when Pi=P1 and a=4 when Pi=P5
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

AS164
$P_{11}s1 = \sum\limits_{i=1}^{4} P_{i1} ~~~~~~ P_{14}s1 = \sum\limits_{j=1}^{6} P_{3j} ~~~~~~ P_{14}s2 = \sum\limits_{j=1}^{6} P_{2j}s1$
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.
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

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

$\sum\limits_{i=1}^{9} P_{i} = 32$
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)
AS201 P3> P1> P2> P4

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.

$P_{21j} = \sum\limits_{i=2}^{10} P_{2ij}$
AS208     Pi1= 0

17 ≤ Pi6= max(j) Pij≤ 72 a
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) f

P6j= P7j for j=(13, 14, 15)

AS209
1. Pij< P3j< P5j    for j=(1,...,8)
2. P2j< P4j< P6j    for j=(1,...,9)
1. Within Part II:
1. The pendant values plus their subsidiary values have the same rank order in groups 3 and 5 except for a reversal between pendants 3 and 4. That is:

For i=(3,5) with subsidiaries:
Pi8> Pi6> Pi7> Pi4> Pi5> Pi2> Pi1
Pi8> Pi6> Pi7> Pi3> Pi5> Pi2> Pi1

Also, with or without subsidiaries, for i=(3,5)
Pi8> Pi6> Pi7> Pi4> Pi5
Pi8> Pi6> Pi7> Pi3> Pi2> Pi1

2. The pendant values have the same rank order in groups 4 and 6 except for a reversal between pendants 1 and 5. That is:

For i=(4,6)
Pi9> Pi3> Pi4> Pi7> Pi5> Pi6> Pi8> Pi2
Pi9> Pi3> Pi4> Pi7> Pi1> Pi6> Pi8> Pi2

3. Fo i=(1,...,8) withsubsidiaries P5j/P3j is between 1.46 and 1.70. The ratios for j=(2,5,6,7) with subsidiaries are very close to each other and very close to 3/2: That is:

P5j/P3j= 1.5 ± 0.7% for j=(2,5,6,7) (with subsidiaries)

Also, the ratios for j=(2,5) differ from each other by only 0.02%. i
4. For j=(1,...,8) P6j/P4j is between 1.36 and 1.73.
(For the exception j=9, P69/P49= 2.900. ) The ratios for j=(3,4) are very close to each other, namely both are 1.4731 ± 0.1 %.
2. Within group 5, the ratios Ps6/Ps7 and Ps4/Ps1 are the same to within 0.03%. This is of interest because they are remarkably close to √2 . In fact, Ps4/Ps1 = 379/268 = √2 - 0.0024%. There is no reason to hypothesize that the ratios of these pendant values or √2 were of any importance to the khipumaker. For us, however, it suggests an excellent approximation for √/2.
1. The number of pendant values that are multiples of P56' and P57' suggest their values as some kind of units:

P14 = 4 P56'
P48 = 12 P56' = 3 P14
P52 = 18 P56'
Where P56' = 45, so these are 4,12,18 times 45

P55' = 5 P57'
P35 = 11 P57'
P42 = 13 P57' = 2 P25
P46 = 4 P55' = 20 P57'
Where P57' = 34, so these are 5,11,13,20 times 34. These are also 2,10,22,26,40 times 16 and, in addition P25 = 13*17, P24=19*17

2. The number 17 is prominent in that it is a factor of about 12% of the pendant values. (It and two multiples of it, 34 and 85, also appear on subsidiaries but it is not prominent among subsidiary values.)
3. Sum cords:
1. The last pendant in each of groups 1, 3, 5 is related to the sum of the first four pendants in its group.
 G1: $P_{18}\; \& \;its\;subsidiaries≈\sum\limits_{j=1}^{4} P_{1j}\; \& \;their\;subsidiaries$ (736 ≈ 726 + 2 broken subsidiaries) G2: $P_{38}\; \& \;its\;subsidiaries≈\sum\limits_{j=1}^{4} P_{3j}\;\&\;their\;subsidiaries$ (2508 ≈ 2511) G3: $P_{58}≈ \sum\limits_{j=1}^{4} P_{5j} + \sum\limits_{j=5'}^{7'} P_{5j}$ (4074 ≈ 4076)
2. The last pendant in group 6 is approximately equal to the sum of the last pendants in groups 3 and 5. By comment a) above, they in turn were also sums.

P69 = P38 + P58 + 2.

3. We have labeled the extra LB pendant just after group 4 as P51' because by color and value it appears related to group 5. That is:

P51' = P52 + P55.

4. Although there seems to be little consistency with position, many pendant values are the sums of two or more other pendant values. Some of the details are included here but, so far, an overall explanation or generalization is lacking.
1. There are altogether thirteen values that are sums of pairs of other values. They are P13, P27, P31, P32, P35, P47, P57 (in two ways), P51', P61, P64, P65, and P67 (in two ways).
2. Four values in groups 1 and 2 can be expressed as the sum of three other values. Then, with only five exceptions, each of the values in groups 3-6 can be expressed as the sum of three other values. Moreover, they have several such expressions. In group 2, one value can be expressed as the sum of three values in four different ways. In groups 3-6, two can be so expressed in two different ways, three in three different ways, one in four different ways, eight in five different ways, three in six different ways, four in seven different ways, and one each in eight, ten, and eleven different ways. Since the summands can in turn be sums, these can be combined to form longer sum chains. As one example we use P61 which has one expression as a sum of two values and five as a sum of three values. Because these sums, the sums can be extended to include six expressions as sums of four values, four expressions as sums of five values, and one expression each as sums of six and seven values.

P61= P55' + P47= P55' + (P15+ P33) = P55' + P15+ (P11+ P27+ P23) = P55' + P15+ P11+ P27+ (P57' + P56' + P28)
= P55' + (P22+ P14+ P31) = P55' + P22+ P14+ (P21+ P16)
= P55' + (P12+ P13+ P31) = P55' + P12+ P13+ (P21+ P16)
= P55' + (P56' + P34+ P16) = (P13) + P34+ P6
= P55' + (P26' + P42+ P23) = (P55') + P26+ P42+ (P57' + P56' + P28)
= P48+ (P57' + P17) = P48+ (P25+ P27) = P48+ P25+ (P11+ P12) = P48+ P25+ (P56' + P12+ P21)
= P18+ P21+ P35= P18+ P21+ (P16+ P28)
3. All values in groups 1 and 2 that are sums can be expressed such that the summands are only from groups 1, 2, and 5' (those three anomalous pendants in group 5).

P11+ P22= P27
P55' + P56' = P13
P56' + P57' + P28= P23
P12+ P21+ P5,4+j' = Pj,5+j    forj=(1,2)
P29= P17+ P18+ P26= P55' + P12+ P15+ P21+ P26+ P28
P29= P56' + P18+ P21+ P23+ P25= P56' + P57' + P11+ P13+ P23+ P25+ P27
P29= P56' + P57' + P12+ P18+ P21+ P22+ P26

4. The values that cannot be expressed as sums of two or three other values are

Pij    for i=(1,2,3);  j=(2,5,8)
Pij    for i=(1,2);  j=(1,4)
and P17, P26, P69·
5. In several cases, alternate values in groups 1 or 2 plus a third value from some other group add to a value in group 6. These cases are:

P21+ P23+ P54= P67
P23+ P25+ P18= P66
P25+ P27+ P48= P61
P27+ P29+ P12= P63
P14+ P16+ P42= P66
P15+ P17+ P55= P63
P15+ P17+ P35= P66

Similar sums add to values in groups 3- 5. These are:

P13+ P15+ P35= P41
P13+ P15+ P55= P54
P15+ P17+ P52= P43
P15+ P17+ P43= P36
P16+ P18+ P51= P36

6. The majority of double and triple sums involve at least one value from groups 1,2, or 5'. This is probably because the values in these groups are generally smaller in magnitude. The distribution of pendant values is:

0-500 501-1000 1001-1500   1501-2500   ≥2501
G1,G2,G5' 18 1 1 - -
G3,G4 3 9 2 3 1
G5,G6 - 5 5 4 3

The sums that are most unlikely to be fortuitous are those involving no group 1,2 or 5' values as summands. They are among the values larger in magnitude (all ≥ 1001) and all are in Part II. They are:

P33= P35+ P53+ P66= P31+ P47+ P61
P49= P41+ P46+ P52
P51' = P52+ P55
P56= P37+ P48+ P53= P46+ P61+ P68
P57= P53+ P62= P37+ P68
P58= P32+ P51' + P64
P64= P48+ P61
P65= P35+ P51
P67= P51+ P62

AS210
1. The 38 values on the khipus are in three ranges: 0 to 11; 22 to 26; and 35 to 50. Those in the highest range are all multiples of 5.
2. Eleven is a pendant value, sum of consecutive pendant values, sum of a group of pendant values, sum of a pendant and its subsidiary, and sum of similarly placed subsidiaries. The summands can be related as follows:

 2 & 2 4 & 1 2 P223 P223' sub of P223' 4 4 2,1 P111 P112 P113 P114 4 & 4 3 P311 sub of P311 8 3 sub of P321 sub of P311 11 P212
AS211
1. By spacing the khipu is separated into three groups of pendants. By color the groups are each divided into two subgroups. The first two groups have 9 pendants per subgroup and the third group has subgroups of 8 and 7 pendants. Therefore the khipu can be described as:

Pijk where
i=(1,2,3) (group by space)
j=(1,2) (subgroup by color)
k=(1,...,9) for i=(1,2) (position in subgroup)
k=(1,...,8) for i=3, j=1 (position in subgroup)
k=(1,...,7) for i=3, j=2 (position in subgroup)
2. Each pendant in group 1 has a subsidiary. Each subsidiary is the color of its host pendant. All subsidiary values are less than 10. No subsidiaries appear in groups 2 and 3.
3. Each group consists of a subgroup of W cords followed by a subgroup of B cords. That is:

i For all i,k:
• Pi1k is W
• Pi2k is B
4. The first pendant in each subgroup contains the maximum value in the subgroup. Depending on the group, the values are in the teens, twenties, or thirties. All other pendant values are less than 10. Specifically:

For j=(1,2)
• P1j1= 22 & 1
• 34 ≤ P2j1≤ 37
• 13 ≤ P3j1≤ 14 & 2
• 9 ≥ Pijk≥ 0 for i=(1,2,3); j=(1,2); k≠(1)
5. With the exception of the first position (and the unknown value due to breakage) the values in the first subgroup of group 2, position by position, are greater han or equal to those in the second subgroup. Also, position by position, they are greater than or equal to those in each subgroup in group 1. That is:

For j=(2,...,8)
• P21j≥ P22j
For j=(1,...,9)
• P21j≥ P11j
• P21j≥ P12j
6. Several values are repeated in the corresponding positions of the subgroups of group 1. They are:

For j=(1,2)
• P1j1= 22 & 1
• P1j2= 0 with subsidiary va1ue 0
• P1j4= 1
• P1j5= 0 with subsidiary va1ue 0
• P1j7= 2 with subsidiary va1ue 0
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
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)
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):

$P_{211} = 4 = \sum\limits_{j=1}^{4} P_{31j}\;\;\;\;(or \sum\limits_{j=1}^{4} P_{33j})$

$P_{212} = 14 = P_{11}$

$P_{213} = 7 = \sum\limits_{j=1}^{4} P_{34j}$

$P_{214} = 37 = \sum\limits_{j=1}^{4} P_{32j}$

$P_{215} = 10 = P_{322}\;subsidiary$

$P_{216} = 4 = \sum\limits_{j=1}^{4} P_{33j}\;\;\;\;(or \sum\limits_{j=1}^{4} P_{31j})$
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,
$P_{8i}=P_{9i}=30\;\;for\;i\;=\;(1,2,3,5)$

$P_{i}=P_{i+1},\;P_{i+2}=P_{i+3}\;\;for\;i=(1,\;4)\;in\;group\;11\;and\;i=1\;in\;group\;13$

$P_{i}+1=P_{i+3},\;P_{i+3}+1=P_{i+6}\;\;for\;i=(1,\;2)$
UR1096
$P_{13}\;≥\;P_{i1}+P_{i2}+P_{i4}+P_{i5}+P_{i6} \;\;\;for\;(i=2,3,4,...,10)$

$P_{13}\;=\;2(P_{i1}+P_{i2}+P_{i4}+P_{i5}+P_{i6})\;\;\;for\;(i=2,3,4)$

$P_{7i}\;=\;\sum\limits_{j=4}^6 P_{ji}\;\;\;for\;(i=1,2,3,4)$
UR1097
$P_{2,2i} = 5\;for\;\;i\;=\;(1,...,9)$

$P_{3,2i} = 5\;for\;\;i\;=\;(1,...,10)$

$P_{2,2i-1} = 1\;for\;\;i\;=\;(4,..,9)$

$P_{3,2i-1} = 1\;for\;\;i\;=\;(4,...,10)$
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;

$\sum\limits_{i=1}^6 P_{i1}s1s1 = 18\&2$
UR1100
$\sum\limits_{i=1}^6 P_{i2} = \sum\limits_{j=1}^{10} P_{2j}$

$\sum\limits_{i=1}^6 P_{i4} = \sum\limits_{j=1}^9 P_{4j}$

$\sum\limits_{i=1}^6 P_{i3} = \sum\limits_{i=1}^6 P_{i7}$

$\sum\limits_{i=1}^6 P_{i5} = \sum\limits_{i=1}^6 P_{i6}$

$\sum\limits_{i=1}^6 P_{i8} = \sum\limits_{i=1}^6 P_{i10}$
UR1103
$P_{2i}\;=\;P_{3i}+P_{3i}s1+P_{4i}\;\;\;for\;i=(7,9)$
UR1104
$P_{14}\;=\;P_{16}+5300\;=\;P_{33}+22000$

$P_{16}\;=\;P_{33}+16700$

$P_{11}\;=\;P_{42}+1200\;=\;P_{44}+1900$

$P_{21}\;=\;P_{35}+600$

$P_{31}\;=\;P_{41}+400$
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.

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
UR1117 P1= P3+ P4

P4= P5+ P6

P6= P9+ P10+ P11

P7= P8+ P9+ P10+ P11

$P_{2} = \sum\limits_{i=4}^{11} P_{i}^2$
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%
UR1126
$\sum\limits_{i=1}^6 P_{i1}=\sum\limits_{i=1}^6 P_{i2}=\sum\limits_{i=1}^6 P_{i3}= 250$
UR1131
$\sum\limits_{j=1}^{7} P_{1j3} = \sum\limits_{j=1}^{7} P_{1j4}$

$\sum\limits_{j=1}^{7} P_{1j2} = \sum\limits_{j=1}^{7} P_{2j4}$

$\sum\limits_{j=1}^{7} P_{1j6} = \sum\limits_{j=1}^{7} P_{2j2}$
UR1135 Pi7> Pi6> Pi8> Pi5     for i=(1,2,3)
UR1136
$\sum\limits_{i=1}^{10} (P_{i4}+P_{i5}) = 433$

$\sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j}$

$\sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j} = \frac{1}{2} \sum\limits_{j=1}^{9} P_{7j}$

$\sum\limits_{j=1}^{9} (P_{4j} + P_{5j}) = \sum\limits_{j=1}^{9} P_{7j}$

$\sum\limits_{j=2}^{9} P_{6j} = 142$

$\sum\limits_{i=6}^{10} P_{i2} = \sum\limits_{i=6}^{10} P_{i3}$

$\sum\limits_{j=1}^{9} (P_{2j}+P_{7j}) = \sum\limits_{j=1}^{9} (P_{3j}+P_{8j})$
UR1138 Pi1= Pi2
Pi3> Pi4> Pi5
Pi36=Pi7
$\frac{P_{i7}}{P_{i1}} \approx 2.4\;\;\;for\;i=(2,3,4,5)$
UR1140
$\sum\limits_{i=1}^{6} P_{i5} = \sum\limits_{i=1}^{6} P_{i6}$

$\sum\limits_{i=7}^{10} P_{i1} = \sum\limits_{i=7}^{10} P_{i2}$

$\sum\limits_{j=1}^{8} P_{1j} = \sum\limits_{j=1}^{8} P_{7j}$
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)
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)

UR1145 P3i+ P3,i+1= P1i           for i=(1,4)

P3i+ P1i= 2 P3,i-1         for i=(4,7)
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.
UR1149
$P_{21j} = \sum\limits_{i=1}^{9} P_{3ij} \;\;\;for\;j=(1,...,5)$

$P_{11j} = \sum\limits_{i=1}^{7} P_{2ij}\;\;\;\;for\;all\;j=(1,...,5)$

$P_{12j} = \sum\limits_{i=8}^{14} P_{2ij}\;\;\;\;for\;all\;j=(1,...,5)$
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

$\sum\limits_{i=6}^{10} P_{3i}\; = \;26$

$\sum\limits_{i=1}^{5} P_{3i}\;\;subsidiaries\; = \;P_{3,12}\;subsidiary$

$\sum\limits_{i=1}^{10} P_{3i}\;\;subsidiaries\; = \;P_{3,11}\;subsidiary + P_{3,12}\;subsidiary = 26$
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)
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
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

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