Problem 1. Consider the following game for two players P1 and P2. P1 picks an integer i between 1 and an upper limit k. P2 tries to determine i by a series of guesses. A guess consists of P2 naming a number j and Pi stating whether j i. The garne is over when P2 guesses j = i. Assuming an optimal strategy is used, how many guesses does P2 need in order to be sure to find i in each of the following cases? Briefly explain each of your answers. 1. k= 1023. 2, k = 1023 and PI tells P2 before the first guess that i is even. 3.k 1023 and PI tells P2 after ansuering the second guess that i is even. 4, k = 2047 and P1 tells P2 before the first guess that i is divisible by 8. 5. k = 255 and PI tells P2 before the first guess that i is a square number, that is, Vi e z+. 6.k-64 and P1 tells P2 before the first guess that i is a power of 2, that is, i=2p for some p > 0