Cyclically symmetric SDP analyses of the disjunction of p variables

The sets of strings given below represent SDP forms of the disjunction of p variables, for p = 3, 5, 7, 11, 13, and 17. Each SDP is cyclically symmetric and has no more than (p-1)/2 negations in any one product. Interpreting a string is simple: each string represents a word in p variables, with a 2 indicating an asserted variable, a 1 indicating a negated variable, and a 0 indicating an absent variable. In each case the full SDP is obtained from the given set of strings as follows: (i) for each given string, include the p-1 other strings obtained by cyclically permuting the indices in that string, and then (ii) also include 222…2. For instance the first set of strings contains only the one string 2 1 0. This represents the following SDP, where ~ represents negation.

x + y + z = x(~y) + y(~z) + z(~x) + xyz

The second set contains 2 1 1 0 0 and 2 2 0 2 1. It represents the following SDP.

a + b + c + d + e =

a(~b)(~c) + b(~c)(~d) + c(~d)(~e) + d(~e)(~a) + e(~a)(~b) + abd(~e) + bce(~a) + cda(~b) + deb(~c) + eac(~d) + abcde.

Note that these forms are not unique; for instance when p = 5 one could replace 2 2 0 2 1 with 2 2 1 2 0. Also, none has been checked for minimality — there may be similar SDP forms with fewer terms. By the way, the SDP form given for p = 7 was used in Lemma 1 (preprint Lemma 3.2) of the paper Non-minimal sums of disjoint products.

2 1 0

2 1 1 0 0
2 2 0 2 1

2 1 1 1 0 0 0
2 2 0 0 2 1 1
2 1 2 0 2 1 0
2 2 2 2 0 2 1

2 1 1 1 1 1 0 0 0 0 0
2 2 0 0 0 0 2 1 0 1 1
2 1 2 0 0 1 2 0 0 1 1
2 1 1 2 0 0 2 0 0 1 1
2 1 1 1 2 0 2 0 0 1 1
2 0 1 2 0 0 1 2 0 1 1
2 1 2 2 0 2 2 0 0 0 1
2 1 2 1 2 2 2 0 1 1 0
2 0 2 2 0 2 1 2 0 1 1
2 0 2 1 2 2 1 2 1 1 1
2 2 2 0 0 0 2 2 1 1 1
2 2 1 0 2 0 2 2 0 1 1
2 1 2 1 2 2 2 1 2 0 0
2 2 0 2 1 2 1 2 1 2 1
2 2 2 1 2 2 0 2 2 0 0
2 2 2 2 0 0 2 2 2 1 1
2 2 2 1 2 1 2 2 2 1 1
2 2 2 2 1 2 2 2 1 2 1

2 1 1 1 1 1 1 0 0 0 0 0 0
2 2 0 0 0 0 0 2 1 0 0 1 1
2 1 2 0 0 0 1 2 0 0 0 1 1
2 1 1 2 0 0 1 2 0 0 1 1 1
2 1 1 1 2 0 0 2 0 0 1 1 1
2 1 1 1 1 2 0 2 0 0 1 1 1
2 0 1 2 0 0 1 1 2 0 1 1 1
2 0 1 1 2 0 0 1 2 1 1 1 1
2 0 2 0 0 2 0 1 2 1 1 1 1
2 2 0 1 0 2 0 1 1 2 1 1 1
2 1 2 1 0 2 0 1 1 2 1 0 1
2 0 0 2 0 0 2 1 1 2 1 1 1
2 1 2 0 2 0 2 2 0 0 0 1 1
2 1 1 2 2 0 2 2 0 0 1 1 0
2 1 2 0 1 2 2 2 0 0 0 1 1
2 1 1 2 1 2 2 2 0 0 1 1 0
2 0 2 0 2 1 2 1 2 1 0 1 1
2 0 1 2 2 1 2 1 2 0 1 0 1
2 0 1 2 0 2 2 1 2 0 1 1 1
2 2 2 0 0 0 0 2 2 1 0 1 1
2 2 1 0 1 2 0 2 2 0 0 1 1
2 2 0 1 2 1 2 1 0 2 0 1 1
2 2 1 1 2 2 0 1 2 2 0 1 1
2 0 2 0 0 2 2 1 2 2 0 0 1
2 2 1 2 2 2 0 2 2 0 0 1 1
2 2 1 2 2 1 2 2 2 0 0 1 0
2 0 2 2 2 1 2 1 2 2 0 0 1
2 2 2 2 0 0 0 2 2 2 1 1 1
2 2 2 1 1 2 0 2 2 2 0 1 1
2 2 2 0 0 2 2 1 2 1 2 0 1
2 1 2 0 2 2 2 1 2 1 2 1 0
2 2 0 2 0 2 1 2 1 2 1 2 1
2 2 2 1 2 0 2 2 2 2 0 1 1
2 2 2 2 2 0 0 2 2 2 2 1 1
2 2 2 2 1 2 0 2 2 2 2 1 1
2 2 2 2 0 2 2 2 1 2 1 2 0
2 2 2 0 2 2 0 2 2 1 2 2 1
2 2 2 1 2 2 2 2 2 2 0 2 0

2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
2 2 1 1 1 1 1 1 1 2 0 0 0 0 0 0 0
2 1 2 1 1 1 1 1 1 1 2 0 0 0 0 0 0
2 0 2 1 1 1 1 1 1 2 0 0 0 0 0 0 0
2 0 1 2 0 1 1 1 1 2 0 0 0 1 1 0 1
2 1 1 1 1 1 2 0 1 1 2 1 0 0 0 0 0
2 0 0 1 2 0 0 1 1 2 1 1 0 0 0 1 1
2 0 0 0 1 2 0 1 1 2 0 0 0 1 1 0 1
2 0 0 0 1 1 2 1 1 2 0 1 0 0 1 1 1
2 1 0 2 0 0 0 0 1 1 2 0 0 1 1 1 1
2 0 0 1 2 0 0 0 1 1 2 1 1 1 1 1 0
2 0 0 1 1 2 0 0 1 1 2 0 1 1 1 0 1
2 1 1 0 0 2 0 0 0 1 1 2 0 0 1 1 1
2 0 2 2 0 1 1 1 1 2 0 0 0 1 1 0 0
2 2 0 0 1 1 2 0 1 1 2 1 1 0 1 1 0
2 1 2 1 1 1 2 0 1 1 2 1 0 0 1 0 0
2 0 0 1 2 0 0 0 1 2 2 1 0 0 0 1 1
2 0 0 2 2 2 0 1 1 2 0 0 0 0 1 0 1
2 0 0 2 2 1 2 1 1 2 0 0 0 0 0 1 1
2 2 0 2 1 2 0 0 1 1 2 0 0 1 0 0 1
2 2 0 2 0 1 1 2 1 0 2 0 0 0 0 1 1
2 1 0 2 0 0 0 2 1 2 2 0 0 1 1 1 1
2 0 0 1 1 2 0 2 1 2 2 0 0 1 1 0 1
2 0 0 1 1 1 2 2 1 2 2 1 0 0 1 1 1
2 2 1 0 2 2 0 0 0 1 1 2 0 0 1 1 1
2 2 0 2 1 1 2 0 1 1 0 2 0 0 1 1 1
2 1 2 2 0 1 2 0 1 1 1 2 0 0 0 0 0
2 0 2 1 2 1 2 0 0 1 1 2 0 1 0 0 0
2 2 0 1 1 1 2 2 0 0 1 2 0 0 1 1 1
2 1 2 1 1 1 2 2 0 0 1 2 0 0 1 0 0
2 0 2 0 2 0 1 1 2 0 0 2 0 1 1 1 1
2 0 1 2 2 1 0 1 2 0 0 2 0 1 1 1 1
2 1 2 0 1 2 0 1 2 0 0 2 0 0 1 1 1
2 1 1 2 0 1 0 2 2 0 1 2 0 0 1 1 1
2 0 2 1 2 0 0 0 1 2 0 2 1 1 1 1 1
2 0 1 1 2 2 0 0 1 2 0 2 0 0 1 1 1
2 0 0 2 1 1 2 0 1 2 1 2 0 0 1 1 1
2 2 0 0 1 0 1 2 0 2 1 2 0 1 1 1 1
2 0 1 0 1 2 0 2 1 2 1 0 1 0 1 2 1
2 0 0 1 2 1 2 1 2 0 0 1 1 1 2 1 1
1 1 1 2 0 2 0 2 1 2 0 0 1 1 1 2 1
2 0 1 0 1 2 0 1 2 2 0 2 0 0 1 1 1
2 0 1 1 2 2 0 0 1 1 2 2 0 0 1 1 1
2 0 1 0 1 2 0 0 2 1 2 2 0 0 1 1 1
2 2 2 0 0 1 1 2 0 0 1 1 2 0 0 1 1
2 2 1 2 1 1 1 2 1 0 1 1 2 0 0 1 0
2 0 2 1 1 2 1 2 0 0 1 1 2 0 0 1 1
2 0 2 2 0 0 1 1 2 0 0 1 2 1 1 1 1
2 1 0 2 1 1 2 1 0 1 2 1 0 1 1 2 0
2 0 2 1 1 0 2 1 0 1 1 2 0 1 1 2 1
2 0 1 1 2 0 1 2 2 0 1 1 2 0 0 1 1
2 0 1 1 2 0 0 2 1 2 1 1 2 0 0 1 1
2 2 0 0 1 1 2 0 1 1 2 1 2 0 1 0 0
1 2 1 1 0 2 0 0 2 2 1 1 1 1 2 0 1
2 0 0 0 2 1 0 1 2 1 0 2 1 2 1 1 1
2 1 0 2 1 1 2 0 1 1 2 1 0 2 0 0 1
2 2 0 2 2 0 2 0 1 1 2 0 0 0 0 1 0
2 0 1 0 2 2 0 0 2 2 0 2 0 0 1 1 1
2 0 2 1 2 0 2 0 0 1 2 2 0 1 1 1 1
2 0 2 1 2 0 1 2 0 1 2 2 0 1 1 1 1
2 0 1 0 2 2 0 0 2 1 2 2 0 0 1 1 1
2 0 0 2 1 1 2 1 1 2 2 2 0 1 1 1 1
2 0 2 1 0 2 1 1 2 2 1 1 2 0 1 1 1
2 0 2 1 2 1 0 2 1 1 2 1 2 0 0 1 0
2 0 2 0 1 0 2 0 2 1 2 1 2 0 1 1 1
2 1 0 2 0 0 1 2 2 0 2 1 2 1 1 1 1
2 2 0 0 1 2 0 0 2 1 1 2 2 0 1 1 1
2 0 1 1 2 1 1 2 1 2 1 2 2 0 0 1 1
2 2 0 2 1 2 1 1 1 2 0 1 1 2 1 0 1
2 1 0 2 2 0 1 2 0 2 1 1 1 2 0 1 1
2 2 0 1 2 0 0 2 1 1 2 1 1 2 0 1 0
2 1 2 1 2 0 0 2 1 1 2 0 1 2 0 1 1
2 2 0 0 1 2 1 2 1 1 2 1 1 2 1 0 1
2 2 0 1 1 2 0 0 2 1 1 2 1 2 1 1 1
2 1 2 0 0 2 0 1 2 1 0 2 1 1 2 1 1
2 2 0 2 2 2 1 2 1 0 2 0 0 0 1 0 1
2 2 0 2 1 2 0 2 1 2 2 0 0 1 1 0 1
2 2 0 2 0 1 2 2 1 2 2 0 0 1 1 1 1
2 2 2 2 2 1 2 0 0 1 1 2 0 1 0 1 0
2 1 2 2 2 0 2 1 2 0 0 2 0 1 1 0 1
2 0 2 1 2 2 2 1 2 0 1 2 0 0 1 1 1
2 2 1 2 2 1 0 2 2 0 0 2 0 1 1 1 1
2 2 0 2 1 1 2 2 2 0 1 2 0 0 1 1 1
2 1 2 2 1 1 2 2 2 0 1 2 0 0 1 0 1
2 2 0 2 2 0 0 2 1 2 1 2 0 0 1 1 1
2 1 2 2 2 0 0 2 0 2 1 2 1 0 1 1 1
2 1 1 2 2 0 2 2 1 2 1 2 1 0 1 1 1
2 2 0 0 1 2 2 2 0 2 1 2 0 1 1 1 1
2 1 2 0 1 2 2 2 0 2 1 2 1 1 1 1 1
2 1 2 2 2 1 2 1 2 0 0 1 1 1 2 1 1
2 2 2 0 1 2 0 1 2 2 0 2 0 0 1 1 1
2 0 2 1 2 1 2 0 2 2 0 2 0 1 1 1 1
2 0 2 1 2 0 1 2 2 2 0 2 1 1 1 1 1
2 1 1 2 1 2 0 2 2 2 0 2 0 0 1 1 1
2 2 2 0 1 2 0 0 2 1 2 2 0 0 1 1 1
2 1 0 2 2 1 0 2 2 1 2 2 0 1 1 1 1
2 1 2 0 1 2 1 2 2 0 2 2 0 0 1 1 1
2 0 1 1 1 2 0 2 2 2 2 2 0 0 1 1 1
2 0 2 1 2 2 0 2 2 0 1 1 2 0 0 1 1
2 0 1 1 2 2 2 2 2 0 1 1 2 0 1 1 1
2 2 1 2 2 0 0 2 0 2 1 1 2 0 0 0 1
2 0 2 1 2 2 0 2 1 2 1 0 2 0 0 1 1
2 2 0 2 1 0 2 2 0 2 1 1 2 0 1 1 1
2 1 2 2 0 1 2 2 0 2 1 1 2 0 1 1 0
2 0 2 1 2 1 2 2 0 2 1 1 2 0 0 1 1
2 1 1 2 1 0 2 2 2 2 1 1 2 0 1 1 1
2 0 1 1 2 1 2 2 2 2 0 1 2 0 0 1 1
2 0 2 1 2 0 1 1 2 2 2 1 2 1 1 1 1
2 0 2 1 1 2 1 1 2 2 2 1 2 0 1 1 1
2 2 2 0 2 1 1 2 0 1 1 2 2 0 0 1 1
2 0 2 1 2 2 0 1 1 2 0 2 2 0 1 1 1
2 0 2 1 2 1 2 0 1 2 0 2 2 1 0 1 1
2 1 2 1 1 2 1 2 0 2 1 2 2 0 1 1 1
2 0 1 1 2 1 2 0 2 2 0 2 2 1 1 1 1
2 0 1 1 2 1 2 0 2 1 2 2 2 1 0 1 0
2 0 0 1 2 1 1 2 1 2 2 2 2 0 0 1 1
2 1 2 2 2 1 1 1 2 2 0 0 1 2 0 1 1
2 1 2 2 2 0 2 0 1 1 2 1 1 2 0 0 0
2 2 0 0 1 2 2 2 1 1 2 1 1 2 1 0 1
2 1 0 2 1 2 2 2 1 0 2 1 1 2 0 0 1
2 0 2 1 2 2 1 0 2 1 2 0 1 2 0 1 1
2 1 1 2 2 0 0 2 2 1 2 1 1 2 0 1 1
2 2 0 0 1 2 0 2 2 0 2 1 1 2 1 1 1
2 2 0 0 1 2 0 2 1 2 2 1 1 2 1 1 1
2 0 0 2 1 1 2 2 1 2 2 1 0 2 0 1 1
2 0 2 0 2 2 1 1 2 0 1 2 0 2 0 1 1
2 0 2 1 2 1 2 0 0 2 1 2 1 2 1 1 1
2 1 0 2 1 2 2 1 1 2 0 2 1 2 1 1 1
2 2 0 2 1 1 1 2 0 2 1 2 0 2 1 1 1
2 0 1 2 2 1 0 1 2 2 0 2 1 2 0 1 1
2 0 0 2 1 1 2 1 1 2 2 2 0 2 0 1 1
2 1 0 2 1 2 2 1 1 2 0 1 2 2 0 0 1
2 0 0 2 1 2 2 0 1 1 2 1 2 2 0 0 1
2 2 1 2 1 2 0 1 2 0 1 2 1 1 2 0 1
2 0 2 2 1 1 2 1 2 0 1 2 0 1 2 1 1
2 2 0 1 2 0 2 1 1 2 0 2 1 1 2 1 1
2 1 2 2 2 1 2 2 2 1 1 2 0 1 1 1 1
2 1 2 0 1 2 2 2 2 0 2 2 0 0 1 1 1
2 1 0 2 2 1 0 2 2 2 2 2 0 1 1 1 1
2 1 1 2 2 0 2 2 2 2 0 1 2 0 0 1 1
2 0 2 2 2 0 2 0 2 1 2 1 2 1 1 0 1
2 0 2 1 2 2 2 0 2 1 2 1 2 1 1 0 1
2 2 2 2 0 0 1 2 2 0 2 1 2 0 1 1 1
2 2 1 2 2 0 1 2 2 0 2 1 2 1 0 0 1
2 2 1 2 1 2 1 2 2 0 2 1 2 0 0 0 1
2 0 2 1 2 1 2 2 2 0 2 1 2 1 1 1 1
2 0 2 0 1 0 2 2 2 2 2 1 2 0 1 1 1
2 2 2 1 2 1 1 2 1 2 1 2 2 0 0 1 1
2 1 0 2 2 1 2 2 1 2 1 2 2 0 1 0 1
2 1 2 1 1 2 2 2 0 2 1 2 2 0 1 1 1
2 2 2 2 2 0 1 0 2 0 2 1 1 2 0 1 1
2 2 2 1 2 1 0 2 2 0 2 1 1 2 0 1 1
2 2 1 1 2 0 0 2 2 2 2 1 1 2 0 1 1
2 1 0 2 2 0 1 2 2 2 2 1 1 2 0 1 1
2 2 0 2 1 2 2 0 2 1 1 2 1 2 0 1 1
2 2 1 2 2 1 0 2 2 0 1 2 0 2 1 1 1
2 0 2 0 2 1 2 2 2 1 1 2 0 2 1 1 1
2 2 2 1 2 2 0 0 1 2 0 2 1 2 0 1 1
2 1 2 1 2 2 2 0 1 2 0 2 1 2 1 0 1
2 2 2 1 2 1 1 2 0 2 1 2 1 2 0 1 1
2 1 2 0 1 2 2 2 0 2 1 2 1 2 1 0 1
2 1 2 1 2 0 1 2 2 2 1 2 1 2 1 1 1
2 2 0 0 1 2 0 2 2 2 1 2 1 2 1 0 1
2 2 2 1 2 0 0 2 1 1 2 2 1 2 1 1 1
2 0 2 1 2 1 2 0 1 2 2 2 1 2 1 1 1
2 1 2 2 0 0 0 2 1 2 2 2 0 2 0 1 1
2 0 2 2 0 2 1 1 2 2 0 1 2 2 0 1 1
2 2 0 2 1 2 1 2 1 0 2 0 2 2 0 0 1
2 0 2 2 0 2 1 1 2 1 2 1 2 2 0 1 1
2 0 2 1 2 2 1 1 2 1 2 0 2 2 0 1 1
2 0 2 1 2 1 2 0 1 2 0 2 2 2 0 1 1
2 1 0 2 1 2 2 1 1 2 0 2 2 2 1 0 1
2 0 2 1 2 0 0 2 1 1 2 2 2 2 0 1 0
2 1 2 2 2 1 2 1 2 0 0 2 0 1 2 1 1
2 2 1 2 1 2 0 2 2 0 1 2 0 1 2 0 1
2 0 2 2 1 1 2 2 2 1 1 2 0 1 2 1 1
2 2 2 2 1 0 2 1 1 2 1 2 1 1 2 1 1
2 2 1 2 1 2 2 0 1 2 1 2 0 1 2 1 1
2 2 2 1 2 1 0 2 0 2 1 2 1 1 2 1 1
2 2 0 1 2 2 0 2 0 2 1 2 1 1 2 1 1
2 0 2 2 1 2 1 0 2 2 0 2 1 1 2 1 1
2 1 2 0 2 2 0 1 2 2 0 2 1 1 2 1 1
2 0 2 1 2 1 2 1 2 2 0 2 0 1 2 1 1
2 0 2 1 2 1 2 1 2 1 2 2 0 1 2 1 1
2 1 2 0 0 2 0 2 2 1 2 2 1 1 2 1 0
2 2 0 2 1 2 0 2 0 2 1 1 2 0 2 1 1
2 2 1 2 1 0 2 1 2 1 0 2 2 1 2 1 1
2 0 2 1 2 2 0 1 2 2 1 1 2 0 2 1 1
2 0 2 0 2 1 2 1 2 2 1 1 2 1 2 1 1
2 1 2 2 1 0 2 0 2 1 2 0 2 1 2 1 1
2 2 0 2 2 2 2 2 1 2 2 0 0 0 1 0 1
2 2 2 2 2 2 2 1 2 0 0 2 0 0 0 1 1
2 2 2 2 2 1 2 0 2 2 0 2 0 1 1 0 1
2 2 2 2 2 0 1 2 2 2 0 2 0 1 1 1 1
2 0 2 1 2 2 2 2 2 2 0 2 0 0 1 1 1
2 2 2 2 2 1 2 0 2 1 2 2 0 1 1 1 1
2 2 2 2 2 0 1 2 2 1 2 2 0 1 1 1 1
2 1 2 2 2 2 0 2 2 1 2 2 0 0 1 1 1
2 0 2 1 2 2 2 1 2 2 2 2 0 0 1 1 1
2 2 2 0 1 2 0 2 2 2 2 2 0 0 1 1 1
2 2 1 2 1 2 0 2 2 2 2 2 0 0 1 1 1
2 2 2 2 2 0 2 2 0 2 1 1 2 1 0 1 1
2 1 2 2 0 2 2 2 2 2 1 1 2 0 1 0 1
2 0 0 1 2 2 2 2 2 2 2 1 2 0 1 1 1
2 1 1 2 2 2 2 2 1 2 1 2 2 0 0 1 1
2 2 2 1 2 0 1 2 2 2 0 2 2 1 1 1 1
2 1 2 1 2 2 1 2 2 2 0 2 2 0 0 1 1
2 0 2 1 2 2 0 2 1 2 2 2 2 0 0 1 1
2 1 2 1 2 1 1 2 2 2 2 2 2 1 0 1 1
2 2 2 0 2 2 2 0 2 0 2 1 1 2 0 1 1
2 1 2 2 2 2 0 2 2 1 2 1 1 2 0 1 1
2 2 0 2 2 1 2 2 2 0 2 1 1 2 0 1 1
2 2 0 2 2 1 2 2 1 2 2 1 0 2 0 1 1
2 2 2 1 2 2 1 0 2 2 2 1 1 2 0 1 1
2 1 2 1 2 2 2 1 2 2 2 1 1 2 0 0 0
2 1 2 2 2 2 2 1 2 0 1 2 0 2 0 1 1
2 2 2 2 2 1 0 0 2 2 1 2 1 2 1 1 1
2 1 2 2 2 1 2 1 2 2 1 2 1 2 1 1 0
2 0 2 2 2 2 1 1 2 0 2 2 0 2 0 1 1
2 2 2 1 2 1 1 2 1 2 2 2 1 2 1 1 1
2 2 0 2 0 1 2 2 1 2 2 2 0 2 0 0 1
2 0 2 0 2 1 2 1 2 2 2 2 1 2 0 1 0
2 2 2 0 2 1 2 2 2 1 0 1 2 2 0 0 1
2 2 1 2 2 0 1 2 2 0 2 1 2 2 0 0 1
2 1 0 2 2 1 2 2 1 2 2 1 2 2 0 1 1
2 0 0 2 1 2 2 2 1 2 2 1 2 2 0 0 1
2 2 0 2 1 2 1 2 1 2 1 2 2 2 0 0 1
2 2 2 1 2 2 2 1 2 0 0 2 0 1 2 1 1
2 1 2 2 2 1 2 2 2 1 1 2 0 1 2 1 1
2 1 2 2 2 2 2 1 1 2 0 2 1 1 2 1 0
2 2 1 2 2 1 2 2 0 2 1 2 0 1 2 0 1
2 2 1 2 2 0 1 2 2 2 1 2 1 0 2 1 1
2 2 2 0 1 2 2 0 2 0 2 2 1 1 2 1 1
2 2 2 2 1 0 2 0 1 2 2 2 1 1 2 1 0
2 2 1 2 1 2 2 1 1 2 2 2 0 1 2 1 0
2 2 0 1 2 2 0 2 1 2 2 2 1 1 2 1 1
2 1 2 1 1 2 2 0 2 2 2 2 1 0 2 0 1
2 1 2 1 1 2 1 2 2 2 2 2 1 1 2 1 1
2 1 2 2 2 1 2 2 0 2 1 1 2 1 2 1 1
2 2 1 2 2 2 1 2 1 1 2 0 2 0 2 1 0
2 2 0 2 1 2 2 0 1 2 2 1 2 1 2 1 1
2 2 0 2 1 2 1 2 1 2 2 1 2 1 2 1 1
2 0 2 1 2 2 1 1 2 2 2 1 2 1 2 1 1
2 0 2 0 2 1 2 1 2 2 2 1 2 1 2 1 1
2 2 2 0 1 2 2 1 2 1 1 2 2 1 2 1 1
2 1 2 2 1 2 2 1 2 1 1 2 2 1 2 1 1
2 2 2 0 1 2 0 2 2 0 1 2 2 1 2 1 1
2 2 0 2 1 2 1 2 1 2 1 2 2 1 2 0 1
2 2 1 1 2 2 0 2 1 2 0 2 2 0 2 1 1
2 2 0 2 0 2 0 2 1 2 2 0 1 2 2 1 1
2 2 2 1 1 2 0 2 2 1 1 2 0 2 2 1 1
2 0 2 0 2 1 2 2 2 1 1 2 0 2 2 1 1
2 2 1 2 0 2 0 2 1 2 1 2 1 2 2 1 1
2 1 2 2 0 2 0 2 0 2 1 2 1 2 2 1 1
2 1 2 1 2 2 0 2 1 2 1 2 1 2 2 1 1
2 0 2 0 2 1 2 1 2 2 1 2 1 2 2 1 0
2 0 2 1 2 1 2 1 2 1 2 2 1 2 2 1 1
2 1 2 2 2 2 0 2 2 2 2 2 0 0 0 1 1
2 2 0 2 2 0 2 2 2 2 2 1 2 1 0 1 1
2 1 2 2 2 0 2 2 2 2 2 1 2 1 1 1 1
2 1 2 2 2 1 2 2 2 2 1 2 2 1 1 0 1
2 2 1 2 2 2 2 2 2 0 2 1 1 2 0 1 1
2 2 2 0 2 2 1 2 2 2 0 2 0 2 0 0 1
2 2 2 1 2 2 2 0 2 1 2 2 0 2 0 1 1
2 2 2 2 2 1 1 1 2 2 2 2 1 2 0 1 1
2 2 2 1 2 2 1 1 2 2 2 2 0 2 0 1 1
2 1 2 1 2 2 1 2 2 2 2 2 1 2 1 1 1
2 2 2 2 2 0 2 2 0 2 1 1 2 2 1 1 1
2 2 2 0 2 2 2 0 2 1 2 1 2 2 0 0 1
2 1 2 2 2 2 2 0 2 1 2 1 2 2 0 1 1
2 1 0 2 2 2 2 2 1 2 2 1 2 2 0 0 1
2 0 2 0 2 1 2 2 2 2 2 1 2 2 0 1 1
2 2 2 0 2 1 1 2 2 2 1 2 2 2 1 1 1
2 2 0 2 1 2 1 2 2 2 1 2 2 2 0 0 1
2 0 2 0 2 1 2 2 2 2 1 2 2 2 1 1 1
2 2 0 2 1 2 2 2 1 0 2 2 2 2 0 1 1
2 0 2 1 2 2 1 2 2 1 2 2 2 2 0 1 1
2 2 1 2 1 2 2 1 1 2 2 2 2 2 1 1 1
2 1 2 1 2 2 1 1 2 2 2 2 2 2 0 1 1
2 2 2 2 1 2 2 2 0 2 1 2 1 1 2 1 1
2 2 2 2 1 2 2 1 2 2 1 2 1 0 2 0 1
2 0 2 2 2 2 0 2 2 2 1 2 1 1 2 1 1
2 2 2 2 2 1 2 1 2 1 2 2 0 1 2 1 1
2 2 2 0 2 2 0 2 2 0 2 2 0 1 2 1 1
2 2 2 2 2 2 1 2 0 2 1 0 2 1 2 0 1
2 1 2 2 2 2 2 1 2 2 1 1 2 1 2 1 1
2 2 2 2 0 2 2 0 2 1 2 1 2 1 2 1 1
2 2 2 1 2 2 2 0 2 1 2 1 2 1 2 0 1
2 2 2 0 2 1 2 2 2 1 2 1 2 1 2 1 1
2 1 2 2 2 2 2 0 1 2 2 0 2 1 2 1 1
2 0 2 2 2 2 1 2 1 2 2 1 2 1 2 1 0
2 2 1 2 2 2 2 1 2 1 0 2 2 0 2 1 1
2 1 2 2 2 2 2 1 2 0 0 2 2 1 2 1 1
2 2 0 2 2 2 2 1 1 2 0 2 2 1 2 1 1
2 1 2 2 2 2 2 1 1 2 1 2 2 1 2 1 1
2 1 2 2 2 1 2 2 0 2 1 2 2 1 2 1 1
2 1 2 1 2 2 2 1 2 2 0 2 2 0 2 1 1
2 2 2 1 2 0 0 2 2 2 1 2 2 1 2 1 1
2 2 1 2 2 1 1 2 2 2 0 2 2 1 2 1 1
2 1 2 1 2 0 2 2 2 2 1 2 2 1 2 1 1
2 2 2 2 1 1 2 1 2 1 2 2 2 1 2 1 1
2 1 2 1 2 2 2 0 2 1 2 2 2 1 2 1 1
2 2 1 2 2 0 0 2 2 1 2 2 2 1 2 1 1
2 1 2 0 2 2 1 2 2 1 2 2 2 1 2 1 1
2 2 2 1 2 1 2 2 0 2 1 2 1 2 2 1 1
2 2 1 2 2 2 0 1 2 2 0 2 1 2 2 1 1
2 2 2 1 1 2 2 0 2 2 0 2 1 2 2 1 1
2 2 1 2 1 2 2 0 2 2 0 2 1 2 2 0 1
2 1 2 2 1 2 2 1 2 2 1 2 1 2 2 0 1
2 2 1 2 1 2 1 2 2 2 0 2 1 2 2 0 1
2 1 2 1 2 2 0 2 1 2 2 2 1 2 2 1 1
2 0 2 2 1 2 2 0 2 1 2 1 2 2 2 1 1
2 2 2 0 2 2 0 2 1 2 1 2 1 2 0 2 1
2 2 2 1 2 1 2 1 2 2 0 2 1 2 1 2 1
2 2 2 2 2 2 2 2 2 2 0 2 0 0 1 1 1
2 2 2 2 2 2 2 2 2 1 2 2 0 0 1 1 1
2 2 2 2 2 2 1 2 2 1 2 2 0 2 0 1 1
2 2 1 2 2 0 2 2 2 2 2 1 2 2 0 0 1
2 2 1 2 2 2 2 2 2 2 1 2 1 0 2 1 1
2 1 2 2 2 2 2 2 2 2 1 1 2 0 2 0 1
2 2 0 2 2 2 2 2 1 2 2 0 2 0 2 1 1
2 2 2 1 2 2 2 0 2 2 2 1 2 0 2 0 1
2 2 2 1 2 1 2 2 2 2 2 1 2 1 2 1 1
2 2 2 0 1 2 2 2 2 2 2 0 2 0 2 1 1
2 2 2 2 0 2 2 2 1 2 1 2 2 1 2 1 1
2 2 2 2 2 1 2 1 2 2 0 2 2 1 2 1 1
2 2 2 2 2 2 1 2 1 1 2 2 2 0 2 1 0
2 2 2 1 2 2 2 2 1 1 2 2 2 1 2 1 1
2 2 2 2 1 2 2 0 2 1 2 2 2 0 2 1 0
2 2 2 2 1 1 2 2 2 1 2 2 2 1 2 1 1
2 2 1 2 1 2 2 2 2 1 2 2 2 0 2 1 1
2 2 2 2 1 2 2 0 1 2 2 2 2 1 2 1 0
2 1 2 2 2 2 1 2 1 2 2 2 2 1 2 0 0
2 2 1 2 1 2 2 2 1 2 2 2 2 1 2 1 1
2 1 2 2 2 1 0 2 2 2 2 2 2 1 2 1 1
2 1 2 1 2 1 2 2 2 2 2 2 2 1 2 1 1
2 2 2 2 2 2 0 2 1 2 1 2 0 2 2 1 1
2 2 2 1 2 2 2 0 2 2 0 2 0 2 2 1 1
2 2 1 2 2 1 2 2 2 2 1 2 1 2 2 1 1
2 2 2 1 2 2 2 2 1 1 2 2 1 2 2 1 0
2 2 1 2 2 2 0 2 2 1 2 2 0 2 2 1 1
2 1 2 2 2 2 0 2 2 1 2 2 0 2 2 1 1
2 0 2 2 2 1 2 2 2 2 1 1 2 2 2 1 1
2 2 2 2 0 2 1 2 2 0 2 1 2 2 2 1 1
2 2 0 2 2 2 2 1 1 2 2 1 2 2 2 0 1
2 2 0 2 2 2 1 2 1 2 2 0 2 2 2 0 1
2 1 2 1 2 2 2 0 2 2 2 1 2 2 2 1 1
2 1 2 2 2 1 2 2 0 2 1 2 2 2 2 1 1
2 2 0 2 1 2 2 2 0 2 1 2 2 2 2 1 1
2 0 2 1 2 1 2 2 2 2 1 2 2 2 2 1 1
2 2 1 2 1 2 2 1 2 1 2 2 2 2 2 0 1
2 1 2 2 1 2 2 0 2 1 2 2 2 2 2 1 1
2 0 2 2 1 2 1 2 2 1 2 2 2 2 2 1 1
2 2 2 2 2 1 2 1 2 1 2 2 1 2 1 2 1
2 2 2 1 2 1 2 2 0 2 2 2 0 2 1 2 1
2 2 2 2 1 2 1 2 2 1 2 1 2 2 0 2 1
2 0 2 2 2 2 2 2 2 2 2 1 2 2 0 0 1
2 1 2 2 2 2 2 2 2 1 2 2 2 1 2 1 1
2 2 1 2 2 2 2 0 2 2 2 2 2 1 2 0 1
2 2 1 2 2 2 1 2 2 2 2 2 2 1 2 0 1
2 2 1 2 2 1 2 2 2 2 2 2 2 1 2 1 1
2 2 1 2 1 2 2 2 2 2 2 2 2 0 2 1 1
2 1 2 2 1 2 2 2 2 2 2 2 2 0 2 1 1
2 1 2 1 2 2 2 2 2 2 2 2 2 0 2 1 1
2 2 2 2 0 2 2 2 2 2 0 2 1 2 2 0 1
2 2 2 2 1 2 2 1 2 2 2 2 1 2 2 0 0
2 2 1 2 2 2 0 2 2 2 2 2 0 2 2 1 1
2 2 0 2 2 2 2 1 2 2 2 1 2 2 2 1 1
2 1 2 1 2 2 2 2 2 2 1 2 2 2 2 1 1
2 2 0 2 2 2 2 2 2 2 0 2 0 2 1 2 1
2 2 1 2 2 2 2 2 2 1 2 2 0 2 1 2 1
2 2 2 1 2 2 2 2 1 2 2 2 1 2 1 2 1
2 2 2 2 1 2 1 2 2 2 2 2 0 2 1 2 1
2 2 2 1 2 2 2 0 2 2 2 1 2 2 1 2 1
2 2 2 1 2 2 2 2 1 2 1 2 2 2 1 2 1
2 2 1 2 2 1 2 2 2 2 0 2 2 2 1 2 1
2 2 2 1 2 2 1 2 2 1 2 2 2 2 1 2 1
2 2 2 1 2 2 1 2 2 2 0 2 2 1 2 2 1
2 2 2 2 2 2 2 2 2 2 1 2 2 0 2 1 1
2 2 2 2 2 2 2 2 2 1 2 2 0 2 2 1 1
2 2 2 2 2 1 2 2 2 2 2 2 0 2 1 2 1
2 2 2 2 2 2 1 2 2 2 2 1 2 2 0 2 1
2 2 2 2 1 2 2 2 2 2 2 1 2 2 1 2 1
2 2 2 1 2 2 2 1 2 2 2 2 2 2 0 2 1
2 2 2 2 2 2 0 2 2 2 2 2 2 1 2 2 1
2 2 2 2 2 1 2 2 2 2 2 2 2 0 2 2 0
2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1