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Fic. 6. a) In this loony endgame, after taking the two boxes of the 2-chain at the bottom, Player B has to offer a (long) chain or a loop, and choses to offer the 3-chain on the left. b) Player A, who is in control, only takes one box of the 3-chain.

Figure 6 Fic. 6. a) In this loony endgame, after taking the two boxes of the 2-chain at the bottom, Player B has to offer a (long) chain or a loop, and choses to offer the 3-chain on the left. b) Player A, who is in control, only takes one box of the 3-chain.

Related Figures (25)

Fic. 2. Cram: Computing the nim-value of the 2x3 rectangle.
Fic. 2. Cram: Computing the nim-value of the 2x3 rectangle.
Fic. 3. Computing the nim-value of some Chomp positions.
Fic. 3. Computing the nim-value of some Chomp positions.
Fic. 5. The dual Strings-and-Coins position of the game in Figure 4. Fic. 4. A Dots-and-Boxes endgame where any move will concede a box to the opponent.
Fic. 5. The dual Strings-and-Coins position of the game in Figure 4. Fic. 4. A Dots-and-Boxes endgame where any move will concede a box to the opponent.
Fic. 7. Player B offers a loop, then Player A declines the last four boxes.
Fic. 7. Player B offers a loop, then Player A declines the last four boxes.
Fic. 10. Computing the nim-value of a game.
Fic. 10. Computing the nim-value of a game.
while if k is odd then In particular, if all the short chains are of the same length J, then x = y = lk/2 if k is even, and # = I(k—1)/2 and y=I(k +1)/2 (soy—a=l) if k is odd.
while if k is odd then In particular, if all the short chains are of the same length J, then x = y = lk/2 if k is even, and # = I(k—1)/2 and y=I(k +1)/2 (soy—a=l) if k is odd.
Fic. 11. Player B sacrifices two boxes in order to ensure that he is the player in control in the loony endgame.
Fic. 11. Player B sacrifices two boxes in order to ensure that he is the player in control in the loony endgame.
Fic. 12. A pair of earmuffs: after the 3-string (“band”) is offered and its two last boxes are declined, we obtain two 4-loops.
Fic. 12. A pair of earmuffs: after the 3-string (“band”) is offered and its two last boxes are declined, we obtain two 4-loops.
TABLE 1. Effect on v and 7 of removing a string, depending on its stops.
TABLE 1. Effect on v and 7 of removing a string, depending on its stops.
Fic. 13. Dipper: the “handle” (3-string on the right) must be offered first, leaving an independent loop (the “cup”).
Fic. 13. Dipper: the “handle” (3-string on the right) must be offered first, leaving an independent loop (the “cup”).
TABLE 2. Value of d for some single-component loony endgames
TABLE 2. Value of d for some single-component loony endgames
Fic. 14. A pair of shackles: after a 3-string is offered and its two last boxes are declined, we obtain a 4-loop.
Fic. 14. A pair of shackles: after a 3-string is offered and its two last boxes are declined, we obtain a 4-loop.
Fic. 15. In this game d depends solely on what happens on the bottom right corner. Next can force one chain there with the ver- tical dotted move, or no chain at all with the horizontal one.
Fic. 15. In this game d depends solely on what happens on the bottom right corner. Next can force one chain there with the ver- tical dotted move, or no chain at all with the horizontal one.
Fic. 16. Depending on the dashed move chosen by Next in Figure 15, we can end up with d = 5 (top) or d = 4 (bottom).
Fic. 16. Depending on the dashed move chosen by Next in Figure 15, we can end up with d = 5 (top) or d = 4 (bottom).
Fic. 17. Left: an example of Dots-and-Boxes game where Next can only win by playing a loony move. Right: the reply by Player B. Player A will eventually win 8-7, after taking the two declined boxes from the 3-chain and the 2-chain on the bottom right.
Fic. 17. Left: an example of Dots-and-Boxes game where Next can only win by playing a loony move. Right: the reply by Player B. Player A will eventually win 8-7, after taking the two declined boxes from the 3-chain and the 2-chain on the bottom right.
Fic. 18. Left: loony move by Player A leads to a loss. Right: Player B’s reply. Player A will lose 7-8, after taking only the 1- chain and 2 declined boxes from one of the 4-chains.
Fic. 18. Left: loony move by Player A leads to a loss. Right: Player B’s reply. Player A will lose 7-8, after taking only the 1- chain and 2 declined boxes from one of the 4-chains.
Fic. 19. Diagram of changes of state for case 2: Game starts on the state H = «1; K = *1 and must be on the state H = 0; K =0 when reaching the loony endgame. N indicates moves by Next and P moves by Previous; dotted lines indicate moves that may or may not be available.
Fic. 19. Diagram of changes of state for case 2: Game starts on the state H = «1; K = *1 and must be on the state H = 0; K =0 when reaching the loony endgame. N indicates moves by Next and P moves by Previous; dotted lines indicate moves that may or may not be available.
Fic. 20. Unreversed moves in G € P if Next follows Strategies 1 (left) and 2 (right), which allow him to force odd or even d(H*), resp.
Fic. 20. Unreversed moves in G € P if Next follows Strategies 1 (left) and 2 (right), which allow him to force odd or even d(H*), resp.
Fic. 21. Next plays a canonical winning move. d(G*) will be odd.
Fic. 21. Next plays a canonical winning move. d(G*) will be odd.
m depends on the order in which the strings and loops are offered by Left. For instance, when we have earmuffs (Figure 23) we can end up with m = 1 or m = 2, depending on what Left offers first: one of the loops or the band (string), resp., as we explained in Section 2.1 of Chapter 3.
m depends on the order in which the strings and loops are offered by Left. For instance, when we have earmuffs (Figure 23) we can end up with m = 1 or m = 2, depending on what Left offers first: one of the loops or the band (string), resp., as we explained in Section 2.1 of Chapter 3.
Fic. 24. An eight (the captured boxes do not form part of it): when Left offers one of the three strings that go from a 3-joint to the other, what will remain will be a loop.
Fic. 24. An eight (the captured boxes do not form part of it): when Left offers one of the three strings that go from a 3-joint to the other, what will remain will be a loop.
Fic. 25. A T: after offering one of the three legs (strings), what will remain is an independent chain.
Fic. 25. A T: after offering one of the three legs (strings), what will remain is an independent chain.
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