The 23 Monotone symmetric Venn diagrams for n = 7. The G-encoding (RG string) is always shown. For most we also show 1/7 of the diagram, as laid out on a cylinder, and an 0-1 encoding of that layout. ======= !!! Found one !!! ======= 1 {1,2} Has polar symmetry NOTES: - This is "Hamilton" (P2) 10000 10101 11111 11111 11010 10000 A: 1 2 3 4 5 4 3 2 1 4 2 6 2 4 1 2 5 1 3 1 5 2 1 6 5 4 5 6 1 2 1 4 3 6 5 2 D: 1 2 3 4 5 4 3 2 1 4 2 6 2 4 1 2 5 1 3 1 5 2 1 6 5 4 5 6 1 2 1 4 3 6 5 2 C: 1 2 3 4 5 6 1 6 3 2 5 2 3 6 1 2 6 4 6 2 1 6 5 1 3 1 5 6 1 4 5 2 5 4 1 6 B: 1 2 3 4 5 6 1 6 3 2 5 2 3 6 1 2 6 4 6 2 1 6 5 1 3 1 5 6 1 4 5 2 5 4 1 6 ----- --------------------------------- X - - --------- ------------- ----- X X X - - ----- - - ----- - - - X X X X X - - - - - - - - - - - X X X X X - - - - - ----- - - ----- X X X - - ----- ------------- --------- X - ------------------------------------- ======= !!! Found one !!! ======= 5 {4,5} Has polar symmetry NOTES: - This is "Adelaide" (P1) 10000 11010 11111 11111 01101 00100 A: 1 2 3 4 3 2 5 6 4 1 2 1 4 3 1 3 5 6 4 3 4 6 1 3 2 4 1 3 5 2 1 2 5 6 1 3 D: 1 2 3 4 3 2 5 6 4 1 2 1 4 3 1 3 5 6 4 3 4 6 1 3 2 4 1 3 5 2 1 2 5 6 1 3 C: 1 2 3 4 5 2 5 4 1 2 6 5 1 2 3 6 1 6 3 4 1 2 1 6 2 5 2 6 3 4 5 1 6 1 5 2 B: 1 2 3 4 5 2 5 4 1 2 6 5 1 2 3 6 1 6 3 4 1 2 1 6 2 5 2 6 3 4 5 1 6 1 5 2 ----- --------------------------------- X - - - ------------- ------------- X X X - - - - ----- - - ----- - X X X X X - - - - - - - - - - - X X X X X - ----- - - - - ----- - - X X X ------------- - - ------------- - X ----------------- --------------------- ======= !!! Found one !!! ======= 7 {3,6,7,8} NOTES: - This is the diagram discovered by Savage and Winkler in 1992. It was never published, but is refered to in [Gr92b]. - M1. 10000 11010 11111 11111 11001 10000 A: 1 2 3 4 2 4 5 4 2 6 1 2 5 6 5 2 5 4 3 6 1 3 4 3 6 2 1 6 5 1 3 5 3 4 1 6 D: 1 2 3 4 3 2 3 5 2 6 1 6 3 5 4 5 6 4 1 6 2 1 5 3 5 6 1 5 3 4 2 4 1 4 2 6 C: 1 2 3 4 3 2 3 5 6 4 1 6 5 6 4 2 1 4 3 1 6 3 6 5 1 4 1 2 6 5 2 5 3 5 2 4 B: 1 2 3 4 5 4 2 5 1 2 6 1 4 3 4 2 1 4 3 5 6 5 1 5 6 2 1 6 3 5 3 6 3 4 6 2 ----- --------------------------------- X - - - ------------- ------------- X X X - - - - ----- - - ----- - X X X X X - - - - - - - - - - - X X X X X - - - - - ----- ----- - - X X X - - ----- --------------------- - X - ------------------------------------- ======= !!! Found one !!! ======= 11 {11,14,24,34} NOTES: - This is same as the diagram of Figure 6, pg 29, of Grunbaum, Venn Diagrams II, Geombinatorics, 2 (1992) 25-32. - M2 100000 100101 101111 011111 011010 001000 A: 1 2 3 4 3 2 3 4 1 5 1 4 6 2 1 2 6 5 3 5 6 2 1 2 6 2 1 5 3 5 1 4 1 5 3 2 C: 1 2 3 4 5 4 3 2 1 2 3 2 1 4 5 4 1 6 1 4 5 2 1 2 5 6 5 2 5 6 1 4 1 6 3 2 B: 1 2 3 4 5 4 3 2 3 4 1 6 1 4 1 6 3 2 3 6 1 4 1 6 5 4 3 4 5 2 1 2 5 2 1 4 D: 1 2 3 4 5 4 3 6 1 6 3 6 1 4 1 6 5 4 3 4 5 6 5 4 1 2 1 4 1 2 5 6 5 2 1 4 ----- ----------------------------------------- X - - ----------------- ------------- ----- X X X - - ------------- - - ----- - - - X X X X X ----- - ----- - - - - - - - - X X X X X --------- - - - - ----- - - ----- X X X ------------- - - ------------- --------- X ----------------- ----------------------------- ======= !!! Found one !!! ======= 13 {9,13,17,40} NOTES: - Never appeared before? - M3 100000 100110 101111 111011 011010 001000 A: 1 2 3 4 2 4 5 3 1 6 3 1 4 2 1 2 6 4 5 6 4 2 1 2 3 4 3 2 1 5 1 3 1 6 5 2 C: 1 2 3 4 5 3 4 2 1 2 6 4 6 2 1 5 1 6 1 3 5 2 1 2 6 4 2 4 5 6 1 3 6 1 4 2 D: 1 2 3 4 5 3 4 6 2 1 2 4 1 3 1 6 5 3 4 3 6 3 1 4 2 4 1 3 1 2 5 6 2 5 1 3 B: 1 2 3 4 5 4 2 4 1 5 6 5 1 4 1 6 3 2 6 3 1 4 1 6 4 5 3 4 5 2 6 1 6 5 1 4 ----- ----------------------------------------- X - - ----------------- ----- ------------- X X X - - ------------- - - - - ----- - X X X X X - - - ----- - ----- - - - - - X X X X X - ----- - - - --------- - - ----- X X X ------------- - - ------------- --------- X ----------------- ----------------------------- ======= !!! Found one !!! ======= 15 {10,15,26,32} NOTES: - Never appeared before? - M4. 100000 100110 101111 111011 110010 100000 A: 1 2 3 2 4 2 3 2 1 5 1 2 6 2 1 5 1 2 1 3 1 2 1 6 4 5 4 6 1 2 1 3 6 5 4 2 D: 1 2 3 4 5 4 3 2 1 2 6 2 1 2 5 2 1 3 1 2 5 2 1 6 1 4 1 6 1 2 1 4 5 3 6 2 C: 1 2 3 4 5 6 1 6 2 6 5 6 2 6 1 4 1 6 3 6 1 4 1 6 1 2 1 6 1 3 5 4 5 3 1 6 B: 1 2 3 4 5 6 1 6 4 2 3 2 4 6 1 6 5 6 1 6 3 6 1 4 1 6 3 6 1 5 1 2 1 5 1 6 ----- ----------------------------------------- X - - ----------------- ----- ------------- X X X - - ------------- - - - - ----- - X X X X X - - - ----- - ----- - - - - - X X X X X - - - - - ------------- - - ----- X X X - - ----- --------------------- --------- X - --------------------------------------------- ======= !!! Found one !!! ======= 19 {12,16,19,38} NOTES: - Never appeared before? - M5. 100000 100101 110111 011111 011010 010000 A: 1 2 3 2 4 2 5 3 5 2 1 5 1 6 1 5 1 2 3 4 2 4 1 6 4 5 4 1 6 2 1 6 3 5 4 2 D: 1 2 3 4 5 4 2 3 4 1 4 6 1 3 5 3 2 3 5 3 1 5 6 5 1 4 1 6 1 3 1 4 5 6 2 3 B: 1 2 3 4 5 6 1 5 6 2 3 2 5 6 2 1 2 4 1 6 3 6 5 6 3 6 1 3 4 3 1 2 1 4 1 6 C: 1 2 3 4 5 6 1 6 3 6 5 6 4 3 4 6 1 4 1 2 1 4 1 6 3 5 6 5 1 2 5 4 5 1 2 6 ----- ----------------------------------------- X - - ----------------- ------------- ----- X X X - - ----- --------- - ----- - - - X X X X X ----- - - - ----- - - - - - - X X X X X --------- - - - - ----- - - ----- X X X --------- - ----- ------------- --------- X --------- ------------------------------------- ======= !!! Found one !!! ======= 22 {22,35} Has polar symmetry NOTES: - This is "Massey" (P3) 100000 110001 110111 111110 111000 010000 A: 1 2 3 2 4 2 5 2 4 3 1 6 1 3 1 4 1 2 1 6 2 4 5 6 4 1 4 3 2 3 5 6 3 1 5 2 D: 1 2 3 2 4 2 5 2 4 3 1 6 1 3 1 4 1 2 1 6 2 4 5 6 4 1 4 3 2 3 5 6 3 1 5 2 C: 1 2 3 4 5 2 4 1 4 6 3 6 5 2 6 1 5 3 1 3 6 3 4 3 5 3 4 6 1 2 1 6 1 4 1 3 B: 1 2 3 4 5 2 4 1 4 6 3 6 5 2 6 1 5 3 1 3 6 3 4 3 5 3 4 6 1 2 1 6 1 4 1 3 ----- ----------------------------------------- X - - - ----------------------------- ----- X X X - - - - ------------- ----- - - - X X X X X - - - - - ----- - - - ----- - X X X X X - - - - - - - ----- ------------- X X X ----- - - ----- ------------------------- X --------- ------------------------------------- ======= !!! Found one !!! ======= 23 {23,33} Has polar symmetry NOTES: - This is "Victoria" (P4) 100000 110010 110111 101111 001101 000100 A: 1 2 3 4 3 2 5 4 1 2 1 4 3 2 5 4 1 6 5 2 1 2 5 4 1 6 5 6 1 2 1 6 5 4 3 2 D: 1 2 3 4 3 2 5 4 1 2 1 4 3 2 5 4 1 6 5 2 1 2 5 4 1 6 5 6 1 2 1 6 5 4 3 2 C: 1 2 3 4 5 6 1 6 5 4 5 6 3 4 1 6 1 4 5 6 3 4 1 2 3 6 1 6 3 4 1 2 3 2 1 6 B: 1 2 3 4 5 6 1 6 5 4 5 6 3 4 1 6 1 4 5 6 3 4 1 2 3 6 1 6 3 4 1 2 3 2 1 6 ----- ----------------------------------------- X - - - --------------------- ------------- X X X - - - - ------------- - - ----- - X X X X X - - ----- - ----- - - - - - - X X X X X - ------------- - - - - ----- - - X X X --------------------- - - ------------- - X ------------------------- --------------------- ======= !!! Found one !!! ======= 25 {20,25,31,39} NOTES: - Never appeared before? - M6. 100000 101001 111011 011111 001101 001000 A: 1 2 3 2 4 5 2 6 1 6 5 4 3 5 3 6 1 6 2 6 1 6 5 4 5 6 1 5 1 4 1 2 1 3 2 6 B: 1 2 3 4 2 4 5 4 6 4 1 6 5 6 1 4 1 2 1 4 1 3 6 3 5 6 1 4 1 2 6 5 2 3 2 4 D: 1 2 3 4 2 5 2 6 1 2 5 6 2 6 4 6 3 6 1 3 4 3 1 6 1 2 1 6 1 5 3 5 4 3 1 6 C: 1 2 3 4 5 3 5 2 1 2 6 2 1 2 3 4 3 2 1 3 1 4 1 6 1 5 6 2 1 6 5 6 4 3 6 2 ----- ----------------------------------------- X - - --------- --------------------- ----- X X X - - ----- - - ------------- - - - X X X X X ----- - - - - - ----- - - - - X X X X X --------- ----- - - - - ----- - - X X X ----------------- - ----- ------------- - X ----------------- ----------------------------- ======= !!! Found one !!! ======= 27 {21,27,30,36} NOTES: - Never appeared before? - M7. 100000 101010 111011 110111 010101 010000 A: 1 2 3 2 4 2 5 6 1 5 6 4 3 5 3 6 3 1 3 2 1 6 5 4 5 6 5 4 1 4 2 4 1 3 2 6 D: 1 2 3 4 3 5 3 6 1 3 5 6 4 3 4 6 4 2 1 2 4 2 1 6 3 5 6 5 1 5 2 5 4 1 2 6 C: 1 2 3 4 5 2 5 3 5 1 5 6 1 3 2 4 2 3 2 4 1 4 6 4 1 5 6 3 1 6 5 6 4 6 2 3 B: 1 2 3 4 5 2 5 4 5 6 1 6 5 6 1 4 2 3 4 3 1 3 6 3 5 1 6 4 1 6 2 5 2 3 2 4 ----- ----------------------------------------- X - - --------- ------------- ------------- X X X - - ----- - - --------- - ----- - X X X X X - - - - ----- - ----- - - - - X X X X X - ----- - --------- - - ----- - - X X X --------- - ------------- ------------- - X --------- ------------------------------------- ======= !!! Found one !!! ======= 37 {18,28,29,37} NOTES: - Never appeared before? - M8. 100000 101100 111110 110111 000111 000001 A: 1 2 3 4 3 5 3 2 1 5 6 5 4 6 3 2 6 1 3 4 1 4 6 5 3 4 5 2 6 2 1 2 6 5 1 4 D: 1 2 3 4 5 2 4 1 6 4 6 3 5 2 6 2 1 2 5 3 1 3 6 4 5 3 5 4 5 6 1 2 6 4 1 3 C: 1 2 3 4 5 2 4 6 3 6 1 6 3 4 1 2 1 6 5 2 5 4 5 6 1 4 3 4 2 3 5 6 3 1 5 2 B: 1 2 3 4 5 2 5 4 5 3 1 6 3 4 1 2 1 6 2 4 5 6 4 1 3 4 3 2 5 6 3 6 1 6 5 2 ----- ----------------------------------------- X - - --------- ----- --------------------- X X X - - ----- - - - - ----- --------- X X X X X - - - - ----- - - - - - ----- X X X X X - ----- ------------- - - - - - - X X X ----------------------------- ----- - - - X ----------------------------------------- ----- ======= !!! Found one !!! ======= 42 {41,42,46,53} NOTES: - Never appeared before? - M9. 1000000 1000011 1010111 0111101 0111000 0010000 A: 1 2 3 2 4 2 5 6 1 6 2 5 2 4 3 6 3 4 1 4 3 4 6 2 1 6 5 1 5 3 5 4 5 3 1 6 D: 1 2 3 4 3 5 3 6 1 6 5 2 4 2 5 2 6 2 1 6 3 1 4 5 4 6 4 5 1 5 4 3 2 3 1 6 C: 1 2 3 4 3 5 6 2 6 5 1 5 6 5 2 3 1 2 4 1 4 6 4 5 4 6 1 2 1 3 6 3 5 3 4 2 B: 1 2 3 4 5 4 3 4 2 4 1 2 6 1 5 3 5 2 5 3 1 3 5 6 4 6 1 2 1 4 6 5 6 3 6 2 ----- ------------------------------------------------- X - - --------------------------------- ----- ----- X X X - - ------------- ------------- - - - - - X X X X X ----- - ----- - - ----- - ----- - - - X X X X X --------- - - - - - - ------------- ----- X X X ------------- - - ----- ------------------------- X ----------------- ------------------------------------- ======= !!! Found one !!! ======= 47 {45,47,52,56} NOTES: - Never appeared before? - M10. 1000000 1100010 1101011 0011111 0010101 0000001 A: 1 2 3 2 4 3 5 3 2 6 1 6 5 4 3 5 4 6 4 3 1 6 1 2 5 4 5 6 1 5 4 2 1 3 2 6 B: 1 2 3 4 2 5 6 4 1 6 5 6 2 4 1 4 3 5 1 5 6 3 5 6 1 4 1 2 3 6 3 5 2 3 2 4 D: 1 2 3 4 3 5 2 3 2 6 1 2 3 6 2 5 4 6 1 4 5 4 2 6 1 6 3 5 1 5 4 3 5 4 1 6 C: 1 2 3 4 5 3 4 2 4 5 1 2 1 6 3 4 3 2 1 3 4 6 1 5 6 2 1 6 5 6 4 5 3 5 6 2 ----- ------------------------------------------------- X - - - ----------------------------- ------------- X X X - - - - ------------- --------- - ----- - X X X X X ----- ----- - ----- - - ----- - - - - X X X X X ----------------- - - ----- - - ----- - - X X X --------------------- ------------- --------- - - X ------------------------------------------------- ----- ======= !!! Found one !!! ======= 48 {44,48} Has polar symmetry NOTES: - This is "Manawatu" (P6) 1000000 1000101 1101101 0111011 0011010 0010000 A: 1 2 3 4 5 4 3 4 2 4 1 4 6 4 2 1 5 3 5 2 5 1 5 6 5 4 5 6 1 2 1 4 6 3 5 2 D: 1 2 3 4 5 4 3 4 2 4 1 4 6 4 2 1 5 3 5 2 5 1 5 6 5 4 5 6 1 2 1 4 6 3 5 2 C: 1 2 3 4 5 6 1 6 4 2 5 2 4 2 6 2 1 2 3 2 6 1 5 4 5 6 5 1 5 3 5 2 5 3 1 6 B: 1 2 3 4 5 6 1 6 4 2 5 2 4 2 6 2 1 2 3 2 6 1 5 4 5 6 5 1 5 3 5 2 5 3 1 6 ----- ------------------------------------------------- X - - ------------------------- ------------- ----- X X X - - ----- ------------- - - --------- - - X X X X X ----- - - - ----- - ----- - ----- - - X X X X X --------- ----- - - - --------- - - ----- X X X ----------------- - ----- ------------- --------- X ----------------- ------------------------------------- ======= !!! Found one !!! ======= 49 {49,50,51,55} NOTES: - Never appeared before? - M11. 1000000 1100100 1101101 0011111 0010110 0000100 A: 1 2 3 2 4 3 5 6 5 3 2 6 1 4 3 5 4 6 1 6 4 3 1 2 5 4 5 2 1 6 1 5 4 3 2 6 D: 1 2 3 4 5 4 3 2 6 5 1 2 1 5 4 6 5 2 1 3 6 4 1 4 6 5 3 6 3 2 1 3 6 5 4 2 C: 1 2 3 4 5 6 1 5 4 5 3 4 2 6 2 4 5 6 1 3 4 2 3 6 1 6 3 4 1 5 2 3 2 5 1 6 B: 1 2 3 4 5 6 1 6 2 5 4 5 2 6 3 4 1 6 1 4 5 3 4 6 1 2 3 5 1 5 3 4 2 3 2 6 ----- ------------------------------------------------- X - - - --------------------- --------------------- X X X - - - - ------------- - - ------------- - X X X X X ----- ----- - ----- - - - - ----- - - X X X X X ----------------- - - ----- - - - - ----- X X X --------------------- --------- - ----- --------- X --------------------------------- --------------------- ======= !!! Found one !!! ======= 54 {43,54} Has polar symmetry NOTES: - This is "Palmerston North" (P5) 1000000 1100010 1110110 1011101 0001101 0000100 A: 1 2 3 2 4 5 3 2 3 6 1 6 2 5 2 4 3 6 3 4 1 6 3 2 1 6 5 2 1 2 5 3 4 3 1 6 D: 1 2 3 2 4 5 3 2 3 6 1 6 2 5 2 4 3 6 3 4 1 6 3 2 1 6 5 2 1 2 5 3 4 3 1 6 C: 1 2 3 4 3 5 6 2 6 5 1 2 6 3 1 2 4 3 1 3 4 6 5 6 1 2 1 3 6 3 5 4 6 3 6 2 B: 1 2 3 4 3 5 6 2 6 5 1 2 6 3 1 2 4 3 1 3 4 6 5 6 1 2 1 3 6 3 5 4 6 3 6 2 ----- ------------------------------------------------- X - - - ----------------------------- ------------- X X X - - - - ----- ------------- - - --------- X X X X X - - ----- - - - ----- - ----- - ----- X X X X X - ------------- ----- - - - --------- - - X X X ----------------------------- - - ------------- - X --------------------------------- --------------------- ======= !!! Found one !!! ======= 11' NOTES: - Never appeared before? - M12. 100000 010110 111110 111011 001101 000100 ----- ----------------------------------------- X ----- - ------------- ----- ------------- X X X ----- - - ----- - - -- - --------- X X X X X - - - - - - ----- - - --------- X X X X X - ----- ----- - ----- - ----- - - X X X ------------------------- - ------------- - X ------------------------- --------------------- X: 1 2 3 2 4 2 3 5 2 1 5 6 5 1 3 5 2 1 4 3 6 4 6 5 6 4 1 6 4 3 1 6 4 5 2 3 Y: 1 2 3 4 2 3 5 3 2 3 1 2 4 6 5 1 4 5 3 5 4 6 5 1 6 2 6 1 6 4 1 6 5 2 3 4 W: 1 2 3 4 5 6 1 4 5 6 4 5 3 5 4 5 1 4 6 2 3 1 6 3 5 3 6 2 3 1 2 4 2 1 2 6 Z: 1 2 3 4 5 6 1 5 6 5 3 5 6 4 5 1 4 2 4 1 6 4 5 1 3 6 2 3 2 4 2 3 1 2 3 6 ======= !!! Found one !!! ======= 12' NOTES: - Never appeared before? - M13. Y: 1 2 3 2 4 2 3 5 2 1 5 6 5 1 2 3 1 5 4 3 6 4 6 5 6 4 1 6 3 6 4 1 4 5 2 3 W: 1 2 3 4 5 3 5 2 5 3 1 5 4 5 3 1 3 2 6 4 1 6 4 6 3 6 4 2 6 1 2 5 2 1 6 4 X: 1 2 3 4 5 4 3 2 4 1 2 6 2 1 2 3 1 2 4 6 3 6 5 1 5 3 6 5 4 5 6 5 1 6 4 3 Z: 1 2 3 4 5 4 6 1 6 5 4 6 3 6 4 6 1 4 3 5 1 2 5 3 6 3 5 2 3 1 2 4 2 1 2 5 ======= !!! Found one !!! ======= 13' NOTES: - Never appeared before? - M14. Z: 1 2 3 4 2 3 2 5 2 1 5 6 1 5 2 3 1 5 4 3 6 4 5 6 4 6 1 6 3 6 4 1 5 4 2 3 X: 1 2 3 4 5 3 2 5 3 5 1 5 4 5 3 1 2 3 6 4 1 6 4 3 6 4 6 2 6 1 2 5 1 2 6 4 W: 1 2 3 4 5 3 4 2 3 2 1 2 6 1 2 4 1 2 6 3 4 6 5 1 5 4 5 6 5 3 6 5 1 6 3 4 Y: 1 2 3 4 5 3 6 1 6 5 6 3 6 4 3 6 1 3 4 5 1 2 4 5 6 4 5 2 4 2 1 2 3 1 2 5 ======= !!! Found one !!! ======= 20' NOTES: - Never appeared before? - M15. Y: 1 2 3 4 2 3 2 5 2 1 5 6 1 5 3 5 2 1 4 3 6 4 5 6 4 6 1 6 4 3 1 6 5 4 2 3 X: 1 2 3 4 3 2 3 5 2 3 1 2 4 6 5 1 5 4 3 5 4 6 5 6 1 6 2 1 6 4 1 6 2 5 3 4 Z: 1 2 3 4 5 6 1 3 5 6 5 3 5 4 3 5 1 3 6 2 4 1 4 6 5 4 6 2 4 2 1 2 3 1 2 6 W: 1 2 3 4 5 6 1 5 6 4 5 6 5 3 5 1 3 2 1 3 6 3 5 1 4 6 2 4 3 2 4 2 1 2 4 6 ======= !!! Found one !!! ======= 21' NOTES: - Never appeared before? - M16. X: 1 2 3 2 4 2 5 2 6 1 6 5 4 2 4 6 4 2 1 2 4 5 1 3 5 6 4 5 4 3 4 1 3 2 1 5 W: 1 2 3 4 2 5 4 5 1 5 6 1 4 2 1 5 3 2 3 5 6 5 3 5 1 6 2 6 3 1 3 5 3 4 3 2 Y: 1 2 3 4 3 2 5 2 3 2 1 5 4 5 3 1 3 2 3 6 3 4 1 4 3 6 4 2 6 2 1 2 5 1 6 4 Z: 1 2 3 4 5 3 5 2 5 1 2 6 1 3 1 6 2 6 5 6 3 6 4 1 4 3 5 6 5 4 5 6 1 6 5 3 ======= !!! Found one !!! ======= 26' NOTES: - Never appeared before? - M17. Y: 1 2 3 2 4 2 5 6 1 2 5 6 4 2 4 5 4 2 1 2 6 3 4 1 6 5 4 6 4 3 4 1 3 2 1 6 Z: 1 2 3 4 2 5 4 5 1 5 6 1 2 5 4 1 3 2 3 5 6 5 3 5 1 6 3 2 1 6 3 5 3 4 3 2 X: 1 2 3 4 3 5 3 6 1 6 3 5 6 4 5 4 1 4 2 1 6 4 5 1 3 6 3 4 2 4 3 4 1 2 3 6 W: 1 2 3 4 5 2 5 3 5 2 1 2 4 6 5 1 4 3 5 4 5 6 5 1 6 2 1 4 1 2 6 2 5 2 3 4