Modulo 7 Hexamagic Series Lemma If A,B,C,D,E,F are the number of 1,2,3,4,5,6 (mod 7) entries, respectively, in a hexamagic series, then A ≡ 6S1 + 6S2 + 6S3 + 6S4 + 6S5 + 6S6 (mod 7) B ≡ 3S1 + 5S2 + 6S3 + 3S4 + 5S5 + 6S6 (mod 7) C ≡ 2S1 + 3S2 + S3 + 5S4 + 4S5 + 6S6 (mod 7) D ≡ 5S1 + 3S2 + 6S3 + 5S4 + 3S5 + 6S6 (mod 7) E ≡ 4S1 + 5S2 + S3 + 3S4 + 2S5 + 6S6 (mod 7) F ≡ S1 + 6S2 + S3 + 6S4 + S5 + 6S6 (mod 7) Proof [1] A + 2B + 3C + 4D + 5E + 6F ≡ S1 (mod 7) [2] A + 4B + 2C + 2D + 4E + F ≡ S2 (mod 7) [3] A + B + 6C + D + 6E + 6F ≡ S3 (mod 7) [4] A + 2B + 4C + 4D + 2E + F ≡ S4 (mod 7) [5] A + 4B + 5C + 2D + 3E + 6F ≡ S5 (mod 7) [6] A + B + C + D + E + F ≡ S6 (mod 7) Solve [6] for A. [6a] A ≡ S6 + 6B + 6C + 6D + 6E + 6F (mod 7) [1a] S6 + B + 2C + 3D + 4E + 5F ≡ S1 (mod 7) [2a] S6 + 3B + C + D + 3E ≡ S2 (mod 7) [3a] S6 + 5C + 5E + 5F ≡ S3 (mod 7) [4a] S6 + B + 3C + 3D + E ≡ S4 (mod 7) [5a] S6 + 3B + 4C + D + 2E + 5F ≡ S5 (mod 7) Solve [5a] for D. [5b] D ≡ S5 + 6S6 + 4B + 3C + 5E + 2F (mod 7) [6b] A ≡ 6S5 + 2S6 + 2B + 3C + E + 4F (mod 7) [1b] 3S5 + 5S6 + 6B + 4C + 5E + 4F ≡ S1 (mod 7) [2b] S5 + 4C + E + 2F ≡ S2 (mod 7) [3b] S6 + 5C + 5E + 5F ≡ S3 (mod 7) [4b] 3S5 + 5S6 + 6B + 5C + 2E + 6F ≡ S4 (mod 7) Solve [2b] for E. [2c] E ≡ S2 + 6S5 + 3C + 5F (mod 7) [5c] D ≡ 5S2 + 3S5 + 6S6 + 4B + 4C + 6F (mod 7) [6c] A ≡ S2 + 5S5 + 2S6 + 2B + 6C + 2F (mod 7) [1c] 5S2 + 5S5 + 5S6 + 6B + 5C + F ≡ S1 (mod 7) [3c] 5S2 + 2S5 + S6 + 6C + 2F ≡ S3 (mod 7) [4c] 2S2 + S5 + 5S6 + 6B + 4C + 2F ≡ S4 (mod 7) Solve [1c] for F. [1d] F ≡ S1 + 2S2 + 2S5 + 2S6 + B + 2C (mod 7) [2d] E ≡ 5S1 + 4S2 + 2S5 + 3S6 + 5B + 6C (mod 7) [5d] D ≡ 6S1 + 3S2 + S5 + 4S6 + 3B + 2C (mod 7) [6d] A ≡ 2S1 + 5S2 + 2S5 + 6S6 + 4B + 3C (mod 7) [3d] 2S1 + 2S2 + 6S5 + 5S6 + 2B + 3C ≡ S3 (mod 7) [4d] 2S1 + 6S2 + 5S5 + 2S6 + B + C ≡ S4 (mod 7) Solve [4d] for C. [4d] C ≡ 5S1 + S2 + S4 + 2S5 + 5S6 + 6B (mod 7) [1d] F ≡ 4S1 + 4S2 + 2S4 + 6S5 + 5S6 + 6B (mod 7) [2d] E ≡ 3S2 + 6S4 + 5S6 + 6B (mod 7) [5d] D ≡ 2S1 + 5S2 + 2S4 + 5S5 + B (mod 7) [6d] A ≡ 3S1 + S2 + 3S4 + S5 + B (mod 7) [3d] 3S1 + 5S2 + 3S4 + 5S5 + 6S6 + 6B ≡ S3 (mod 7) Multiply [3d] by 6 and solve for B. [3d] B ≡ 3S1 + 5S2 + 6S3 + 3S4 + 5S5 + 6S6 (mod 7) [4d] C ≡ 2S1 + 3S2 + S3 + 5S4 + 4S5 + 6S6 (mod 7) [1d] F ≡ S1 + 6S2 + S3 + 6S4 + S5 + 6S6 (mod 7) [2d] E ≡ 4S1 + 5S2 + S3 + 3S4 + 2S5 + 6S6 (mod 7) [5d] D ≡ 5S1 + 3S2 + 6S3 + 5S4 + 3S5 + 6S6 (mod 7) [6d] A ≡ 6S1 + 6S2 + 6S3 + 6S4 + 6S5 + 6S6 (mod 7)