Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either one, two, or three matchsticks. The player to remove the last matchstick wins the game.(a) Suppose there are only 5 matchsticks left, and it is Fred’s turn. What move should Fred make to guarantee himself victory? Explain your reasoning.(b) Suppose there are 10 matchsticks left, and it is Fred’s turn. What move should Fred make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.)(c) Now start from the beginning of the game. If both players play optimally, who will win?(d) What are the optimal strategies (complete plans of action) for each player?

Which statement about the cost of the options is true? she would save $1,000 by choosing option b. she would save $5,650 by choosing option a. she would save $11,200 by choosing option b. she would save $11,300 by choosing option a.

Select all that apply. what do opponents of minimum wage believe are the results of minimum wage? increases personal income results in job shortages causes unemployment raises prices of goods

Net income and owner's equity for four businesses four different proprietorships, jupiter, mars, saturn, and venus, show the same balance sheet data at the beginning and end of a year. these data, exclusive of the amount of owner's equity, are summarized as follows: total assets total liabilities beginning of the year $550,000 $215,000 end of the year 844,000 320,000 on the basis of the preceding data and the following additional information for the year, determine the net income (or loss) of each company for the year. (hint: first determine the amount of increase or decrease in owner's equity during the year.) jupiter: the owner had made no additional investments in the business and had made no withdrawals from the business. mars: the owner had made no additional investments in the business but had withdrawn $36,000. saturn: the owner had made an additional investment of $60,000 but had made no withdrawals. venus: the owner had made an additional investment of $60,000 and had withdrawn $36,000. jupiter net income $ mars net income $ saturn net income $ venus net income $