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?

Ajax corp's sales last year were $435,000, its operating costs were $362,500, and its interest charges were $12,500. what was the firm's times-interest-earned (tie) ratio? a. 4.72b. 4.97c. 5.23d. 5.51e. 5.80

Asmall town wants to build some new recreational facilities. the proposed facilities include a swimming pool, recreation center, basketball court and baseball field. the town council wants to provide the facilities which will be used by the most people, but they face budget and land limitations. the town has $400,000 and 14 acres of land. the pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. also, the council has only enough flat land to build the basketball court or the baseball field. the daily usage and cost of the facilities (in $1,000) are shown below. which facilities should they build? facilityusagecost ($1,000)landswimming pool4001002recreation center5002003basketball court3001504baseball field2001005

Which of the following would indicate an improvement in a company's financial position, holding other things constant? a. the inventory and total assets turnover ratios both decline.b. the debt ratio increases.c. the profit margin declines.d. the times-interest-earned ratio declines.e. the current and quick ratios both increase.