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?

among other factors, composers started writing dynamic markings in their scores because "new instruments allowed for more dynamic ranges and contrast."

dynamics allude to the loudness or softness of music. dynamics offer an approach to demonstrate articulation in sheet music. they to drive the enthusiastic substance of music through volume and force. maybe you could alter both the volume and the shading profundity on your screen at the same time. we can likewise think about the force side of elements as the quality of a note. a few notes are intended to be played delicately and gently, while others are intended to be played all the more firmly.  

some dynamics markings have a letter 'm,' which remains for mezzo, signifying 'medium.' so a mezzo piano unique stamping resembles this (mp) and intends to play medium calm, while a mezzo specialty dynamic resembles this (mf) and intends to play medium loud.