How many distinct first moves are there in the Nimble position of Figure 1.1.1? (Don't worry about whether these moves are winning or losing; just count the total number of possible first moves.)
2.
Explain why Position 5 in Figure 1.1.2 is Bob-win.
3.
Should Alice have lost the game in Figure 1.2.5? Why or why not?
4.
Consider the Nimble position in Figure 1.4.1, where it is Alice's turn to move. Who is the winner: Alice or Bob? Explain.
Figure1.4.1.Who wins this Nimble position with 4 coins?
5.
Who wins the Nimble position in Figure 1.1.1? Explain. (Hint: your answer to exercise 2 will be helpful.)
6.
Consider a game of Nimble with two coins in position \(\{ r, s \}\text{.}\) Under what conditions is the game position an \(\cN\)-position, and when is it a \(\cP\)-position? Justify your answers by giving a winning strategy in each case.
7.
Create the game tree for the Nimble position \(\{1,1,2 \}\text{.}\) Indicate whether each position in the game tree is an \(\cN\)-position or a \(\cP\)-position.
8.
For each of the game trees in Figure 1.4.2, recursively determine whether each position is an \(\cN\)-position or a \(\cP\)-position. Who wins the root game position, Alice or Bob?
Figure1.4.2.Is the root position an \(\cN\)-position or a \(\cP\)-position ?
