this is problem strategy and choice problem pls help asap

20.5 (Turning Turtles) This is a Nim-like game under a somewhat better disguise than Nimble. Ten turtles are lined up from left to right. Some are on their backs (X) and some right-side up (O). On his or her turn, each player must choose one right-side-up turtle and flip it onto its back (turn O into X). The player may, if he or she wishes, also change the status of one turtle to the left of this chosen one, either from O to X or from X to O. We translate a "Turtle configuration" (i.e., a list of X's and O's) to a Nim configuration as follows: view a right-side-up turtle in position n as a pile of n chips. This gives the Nim configuration for the game. For example, Turtles Nim Piles XOXXOXOXXO (2, 5, 7, 10) OXXOOOXXOX (1, 4, 5, 6, 9) Combinatorial Games 313 (a) Show that the Nim move of removing an entire pile of n chips corre- sponds to the Turning Turtles move of flipping an O in position n to an X. (b) Consider the Nim move of reducing a pile of n chips to a pile of & chips. Since we have a pile of n chips, in Turning Turtles, there must be an O in position n. Flip this O to an X. We now want to add & chips back. If there is an X in position k, we may simply flip this to an O and thereby add a pile of & chips in the Nim configuration. Suppose that there is an O in position k. Prove that flipping this O to an X yields a Nim configuration that is strategically the same as to that obtained by simply reducing the pile of n chips to & chips. Hint: Prove that the Nim configuration with two piles of & is strategically the same as the configuration with both piles of & eliminated