DICE RINK
by Derick Green
(first published in the Games and Puzzles Journal issue 15)

For this game four dice, in two pairs, one pair white and the other coloured, and a 4×4 board are all that is needed. I have no idea who the designer was, but have been told the game was originally produced commercially by Whitman Australia Pty Ltd in the early 1970s, under the uninspired name 'Relate', and may have been based on a mathematical puzzle from the 1950s.

Call the numbers 1 and 2 'low' and regard them as equivalent, the numbers 4 and 5 as 'high' and also equivalent to each other, while the numbers 3 and 6 (easily remembered as being divisible by three) are considered distinct. We call low, high, 3 and 6 the four 'values' of the dice. (The game can alternatively be played with specially prepared dice with coloured instead of numbered faces, say purple, green, yellow and red as the four values. The two purple faces are adjacent, as are the two green faces, and red is adjacent to yellow.)

To start, the first player places one die on one of the four squares on the first rank. The second player responds with a die on one of the four squares on the fourth rank, taking care that its uppermost face does not duplicate the high, low, 3 or 6 shown on the opponent's die. The first player puts down the third die on the first rank, obeying the same restrictions, and the second player places the last die on the fourth rank, ensuring that the four dice now show the four different values, high, low, 3 and 6 on their upper faces. The game is then ready for play. (A simpler alternative opening procedure is for player A to start in b1 and c1 and player B in b4 and c4.)

The game is won when one player has moved both dice to the opponent's starting line. At each turn a player rolls one die through ninety degrees to an adjacent square of the board. The value now shown must not duplicate that on the player's other die but may duplicate that on one of the opponent's dice. If this happens, the opponent must roll that die or, if no legal roll to another square is possible, rotate it on the same square, to show any face different from that on the other die. A player must roll, rather than rotate, if at all possible. This applies even if one of the dice has already reached the opponent's side of the board; the player must move it if the opponent matches the combination on it.

Illustrative Game: Notation: A move is shown by the starting number, direction of movement, and ending number. Set-up: A places 1 at b1, B places 6 at b4, A places 3 at c1, B places 5 at d4, to give the position shown (Figure 1 - Game start).

Play: 1. 1N4 (This move forces B to move his die on d4 and allows A to advance b2 on move 2) 5S4 2. 4N6 6W3 3. 3W5 4W6 4. 6N3 3S2 5. 5N6 6W3 6. 3W2 2S4 7. 2E3 3E6 8. 6N2 4S5 (B's mistake. A better move would have been 4N2 forcing A's 2S6 delaying the win by forcing A to move back.) 9. 2W4 5N4 10. 4N1 and wins. Game end (Figure 2)

The game is not easy to analyse but with some thought it is possible to see a couple of moves ahead. It has been calculated that there are 276,480 possible starting positions, though I would not like to argue the point over this number. Players should always be aware of the number of moves possible by their own and the opponent's dice. For example in Figure 3, where both dice are player A's the die on b2 has four possible moves, but the die on a1 only one. (Figure 3 - Moves available).

Illustrative Endgame: Figure 4 is taken from a postal endgame. A has just moved to b2 with 1. 5N6, B can only make one move 1... 6N2. After this A moves 2. 2N4 then B can only play 2... 5N6. Now A plays 3. 6N2 and B cannot roll the die on d4 so rotates to 5 (with 4 at north). Now A is forced to rotate a4 which he does to 6 with 3 at north. B resigns, since A cannot be prevented from moving 2N1 next turn. (Figure 4 - Endgame)

Despite this example, it is often good practice not to allow the opponent to rotate.

Moving into corners restricts movement and it is often appropriate to move backwards in the middle game to free up the board, although the player who can get one die on the opponent's back rank first can often use this die to sandwich opposing dice, to cut down the opponent's possible moves. Dice Rink is not easy to analyse and tactics and strategy vary with each game because of the variety of opening positions possible.

As regards variants of the game, a friend suggested that the first player has the advantage and put forward the idea that to even things up A should set-up in b1 and c1 only, while B can still use any of his four back-row squares for initial placement. The theory being that A would only have a choice of four possible opening moves but B could have five. I disagree, because with careful placement A can restrict B's opening moves anyway and B gets to place his dice after A and so knows what possible moves A can make. The idea of three dice per side (using all six of the normal dice values) and a bigger board (say 6×6 or 4×6) has also been discussed, but not tried in practice.

Of the 30 games I have on record from the postal group 15 have been wins for the first player, 2 draws agreed on by both players (although I believe both are wins by the first player with good play) and 13 are wins by the second player.

Illustrative Game 1. (Postal play).

The opening position for A in this game is well thought out but the game was perhaps lengthened by his to and fro moves from a1 to b1 and back again. Start: A: b1(top 5, front 4) d1(top 3, front 6) B: b4(top 2, front 4) d4(top 6, front 4) Play: 1. 5W6 6S3 2. 3W2 2E1 3. 2N1 (B has moved in such a way as to trap the die on c4 on the back row until d3 can be moved) 1W2 4. 1E3 (mistake, better would have been 1N5 keeping b4 on the back rank) 3N6 5. 6E5 2S3 6. 3W1 3S5 7. 5W6 (delaying b2`s advance) 6S3 (forced move) 8. 1E3 3N6 9. 6E5 5E1 10. 5W6 6S3 11. 3S2 1S4 12. 6W5 (better would have been 6N3) 4N1 13. 2N3 3W5 14. 5W6 5N6 15. 6E5 (this die is trapped and should have been advanced earlier) 6W4 16. 5E1 1N3 17. 3N5 4W1 (4S5 would have been better) 18. 1N3 3N6 19. 3N6 6W5 (if 17...4S5 had been played 6E2 could now have been moved) 20. 5N4 5E6 (better would have been 5S3) 21. 6E2 1S5 22. rotate 6(2N) 6S3 23. 6W3 3S1 24. 2N4 wins. Final position: A: c4(top 3, front 2) d4(top 4, front 2) B: a3(top 5, front 1) c2(top 1, front 3).

Illustrative Game 2. (Over-the-board play).

Start: A: b1(top 4, front 2) d1(top 3, front 1). B: b4(top 6, front 3) d4 (top 1, front 4) Play: 1.3N6 (this allows an initial advance; B can either move 6E5 or 6S4, both of which allow 4N5) 6S4 2. 4N5 4E5 3. 5E1 1S3 (this may seem to restrict B's movement, but in fact forces an advance; A must move 1S4, 1W5 or 6S3 all of which allow B to advance and keep up the pressure) 4. 6S3 3S6 5. 3W5 (mistake, 1W5 with 1W6 to follow, forcing B to move back) 5W4 6. 5W4 (mistake, 5E3 etc) 4S1 (keeping up the pressure, B is only two moves away from victory, A is five away. B can therefore allow A's single die through.) 7. 1N3 6S4 8. 4E5 4N6 9. 3N6 6N3 (This move sems strange but if 6S4 A moves 5W4 forcing 4W6 and blocking B's second die.) 10. 5E3 3S6 11. 6E2 (mistake, better would have been 6W5) 1S3 12. 3W5 6S4 wins. Final position: A: c1 (top5 front 1) d4(top 2, front 3) B: b1(top 3, front 1) d1(top 4, front 6).