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Connect the dots game

Main Connect-the-Dots page

These tables show the progress of the game for either 8 or 9 nodes.
For 8 nodes, the first player wins, for 9 it's the second player.

Notation:

The two players will be designated P1 and P2, for first and second player.
[square brackets] denote the current move
{curly brackets} denote subgraph cardinalities.
(parentheses) denote nodes in a subgraph. A trailing c means the subgraph is complete.
a,b without parentheses denotes moves by P1. (a,b) denotes moves by P2.
Numbers such as 03 and 11 at the start of lines are the move number.
Moves to completion shows the the total of number moves
in these subgraphs when they are complete (in parentheses),
and the number of remaining moves from this position to reach that state.

Eight Nodes

The winning positions of three subgraphs for P1 are {6 1 1} and {5 2 1},
with 15 and 11 moves required for completion, and the positions which lead to them.

01: 0,1 => (0,1) 2 3 4 5 6 7 = {2 1 1 1 1 1 1}
  • 02: 0,1, (2,3) => (0,1) (2,3) 4 5 6 7 = {2 2 1 1 1 1}

    • 03: 0,1 (2,3) 0,2 => (0,1,2,3) 4 5 6 7 = {4 1 1 1 1}

  • 02: 0,1 (0,2) => (0,1,2) 4 5 6 7 = {3 1 1 1 1}

    • 03: 0,1,(0,2) 1,2 => (0,1,2)c 3 4 5 6 7 = {3 1 1 1 1}

      • 04: 0,1, (0,2), 1,2, (3,4) =>
        04: (0,1,2,3,4) 5 6 7 = {5 1 1 1}

        05: 0,1, (0,2), 1,2, (3,4), 0,3 =>
        05: (0,1,2,3,4) 5 6 7 = {5 1 1 1}
        05: Could also make {5 2 1} here directly.

      • 04: 0,1, (0,2), 1,2, (0,3) =>
        04: (0,1,2,3) 4 5 6 7 = {4 1 1 1 1}

        05: 0,1, (0,2), 1,2, (0,3) 1,3 =>
        05: (0,1,2,3) 4 5 6 7 = {4 1 1 1 1}

        Continues at right.
Continued from left.
    0,1,(0,2),1,2,(0,3),1,3 --- (0,1,2,3) 4 5 6 7 = {4 1 1 1 1}

    • 06: 0,1, (0,2), 1,2, (0,3), 1,3, (4,5), 0,4 =>
      07: (0,1,2,3,4,5) 6 7 = {6 1 1}
      07: Moves to completion (15): 8

    • 06: 0,1, (0,2), 1,2, (0,3), 1,3, (0,4), 0,5 =>
      07: (0,1,2,3,4,5) 6 7 = {6 1 1}

    • 06: 0,1, (0,2), 1,2, (0,3), 1,3, (2,3), 0,4 =>
      07: (0,1,2,3,4) 5 6 7 = {5 1 1 1}

      • 08: (1,4), 0,5 => (0,1,2,3,4,5) 6 7 = {6 1 1}

      • 08: (0,5), 1,4 => (0,1,2,3,4,5) 6 7 = {6 1 1}

      • 08: (5,6), 1,4 => (0,1,2,3,4) (5,6) 7 = {5 2 1}
        09: Moves to completion (11): 2

Nine Nodes

9 nodes: P2 is able to steer the game to a winning position:
C6 fills on odd: {6 2 1} 16 total moves
C5 fills on even: {5 2 2} 12 total moves
C4 fills on even: {4 4 1} 12 total moves and {4 3 2} 10 total moves
Winning positions for P1: (not reached)
C7 fills on odd: {7 1 1} 21 total moves
C5 fills on even: {5 3 1} 13 total moves
C3 fills on odd: {3 3 3} 9 total moves

01: 0,1 + [0,2] [Alternate response of [2,3] not included, since [0,2] leads to win for P2]
01: (0,1,2) 3 4 5 6 7 8 = {3 1 1 1 1 1 1}
03: Possible moves: [0,3] [1,2] [3,4]

0,1, (0,2) ,0,3

03: 0,3 => (0,1,2,3) 4 5 6 7 8 = {4 1 1 1 1 1}

04: (1,2) => (0,1,2,3) 4 5 6 7 8 = {4 1 1 1 1 1}
Merges with paths below.

0,1, (0,2), 1,2

04: (0,3) => (0,1,2,3) 4 5 6 7 8 = {4 1 1 1 1 1}

05: Possible moves: [1,3] [0,4] [4,5]
  • 05: [0,4] => {5 1 1 1 1}

  • 05: [4,5] => {4 2 1 1 1} => {4 2 2 1}

  • 05: [1,3] + [2,3] => (0,1,2,3)c 4 5 6 7 8 = {4 1 1 1 1 1}

    0,1, (0,2), 1,2, (0,3), 1,3, (2,3)

    07: Possible moves: [0,4] [4,5]

    • 07: 4,5 + (6,7) => (0,1,2,3)c (4,5) (6,7) 8 = {4 2 2 1}
      07: (See below for how any other move causes P2 to lose.)

    • 07: 0,4 + (1,4) => (0,1,2,3,4) 5 6 7 8 = {5 1 1 1 1}

      0,1, (0,2), 1,2, (0,3), 1,3, (2,3), 0,4 (1,4)

      09: Possible moves: [2,4] [0,5] [5,6]

      • 09: 0,5 + (6,7) => (0,1,2,3,4,5) (6,7) 8 = {6 2 1}

      • 09: 5,6 + (0,7) => (0,1,2,3,4,7) (5,6) 8 = {6 2 1}

      • 09: 2,4 + (3,4) => (0,1,2,3,4)c 5 6 7 8 = {5 1 1 1 1}

        0,1, (0,2), 1,2, (0,3), 1,3, (2,3), 0,4 (1,4), 2,4 (3,4)

        11: Possible moves: [0,5] [5,6]

        Same as above.

0,1, (0,2), 3,4

04: 1,2 => (0,1,2)c (3,4) 5 6 7 8 = {3 2 1 1 1 1}

0,1, (0,2), 3,4, (1,2)

05: Possible moves: [0,3] [0,5] [3,5] [5,6]
  • 05: 0,3 + (0,4) => (0,1,2,3,4) 5 6 7 8 = {5 1 1 1 1}

  • 05: 0,5 + (6,7) => (0,1,2,5) (3,4) (6 7) 8 = {4 2 2 1}

  • 05: 3,5 + (4,5) => (0,1,2)c (3,4,5)c 6 7 8 = {3 3 1 1 1}

    0,1, (0,2), 3,4, (1,2), 3,6, (4,5)

    07: Possible moves: [0,3] [0,6] [3,6] [6,7]

    • 07: 0,3 + (6,7) => (0,1,2,3,4,5) (6,7) 8 = {6 2 1}

    • 07: 0,6 + (3,7) => (0,1,2,6) (3,4,5,7) 8 = {4 4 1}

    • 07: 3,6 + (0,7) => (0,1,2,7) (3,4,5,6) 8 = {4 4 1}

    • 07: 6,7 + (0,3) => (0,1,2,3,4,5) (6,7) 8 = {6 2 1}

  • 05: 5,6 + (0,7) => (0,1,2,7) (3,4) (5,6) 8 = {4 2 2 1}

    0,1, (0,2), 3,4, (1,2), 5,6, (0,7)

    07: Possible moves: [0,4] [0,8] [4,6] [4,8]

    • 07: 0,4 + (1,4) => (0,1,2,3,4,7) (5,6) 8 = {6 2 1}

    • 07: 0,8 + (1,8) => (0,1,2,7,8) (3,4) (5,6) = {5 2 2}

    • 07: 4,6 + (1,7) => (0,1,2,7) (3,4,5,6) 8 = {4 4 1}

    • 07: 4,8 + (3,8) => (0,1,2,7) (3,4,8) (5,6) = {4 3 2}


Below we have what would follow if, after 0,1,(0,2),1,2,(0,3),1,3,(2,3),4,5, P2 does not use (6,7).
09: Winning posn: 9 P1 {4 3 1 1}
09: 0,1,(0,2),1,2,(0,3),1,3,(2,3),4,5,(4,6) + [0,4]
09: (0,1,2,3,4,5,6) 7 8

09: Winning posn: 9 P1 {5 2 1 1}
09: 0,1,(0,2),1,2,(0,3),1,3,(2,3),4,5,(0,6) + [0,4]
09: (0,1,2,3,4,5,6) 7 8

09: Winning posn: 9 P1 {6 1 1 1}
09: 0,1,(0,2),1,2,(0,3),1,3,(2,3),4,5,(0,4) + [0,6]
09: (0,1,2,3,4,5,6) 7 8