Hi i have few questions over here and need some expert/pro view with approaching the question. The proving one would be an issue to me as well. Hope people would be able to assist me here , because i am basically struggling with it :( thanks

Q2(a) What is "the Aiken-Dueet ice-cream parlour question"? (Please type it out so we don't have to work with graphical text.)

Q2(b) If the numbers are m, m+a

_{1}, m+a

_{2}, ..., m+a

_{n}, how many differences are there? (BTW: All differences will be of the form a

_{j}-a

_{i}, or else just a

_{j} if you're subtracting m from m+a

_{j}. The m's cancel out.) Given that the differences relate to remainders after division, where does this lead?

Q2(c) If she plays at least one game per day, what is her minimum games per week? If, by day "d", she has played g

_{d} games, her running total is g

_{1}, g

_{2}, ..., g

_{77}, because of seven days per week. If she played exactly 21 games in some stretch of days, then one of these values -- g

_{1} + 21, g

_{2} + 21, ..., g

_{77} + 21 -- matches a value from the first list (because it's that day's total, plus 21, which matches another, later, day's total). Where does this lead?

Q3 Working from the definition of an equivalence relation, what have you done? Where are you stuck?

When you reply, please show your work for Q4 and Q5, too. Thanks.