Question

Alice and Bob play an interactive game as follows. Alice is given 5 empty bags and an unlimited supply of balls of 6 different colors. Then they execute the following steps sequentially: (i) Bob selects a number bE 3,4) and tells it to Alice, (ii) Alice fills the five bags with balls and shows the bags to Bob, (iii) Bob selects b distinct colors and tells it to Alice (iv) Alice wins if she can provide Bob with b balls of the colors chosen by Bob in step (iii) by picking at most one ball from each of the five bags Let nb be the minimum number of balls that Alice needs to put in the bags in step that she wins for any choice of b colors made by Bob in step ii such (a) Find n3 (b) Find n4 Please note the following: The "minimum" in the above questions refers to the minimum of the total number of balls in all the five bags. For example, if Bob selects b = 3, then Alice can put one ball of each of the six colors into each of the five bags, that is, she puts a total of 6× 5 = 30 balls into the five bags. Then for any choice of 3 colors by Bob, Alice wins. However, this is certainly not the minimum possible, as Alice can do better; she can remove any one ball from (any) one of the five bags and again she urns (for any choice of 3 colors by Bob) with less number of balls uhich is 30-1 = 29 in this case. Therefore, clearly n3 29.

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Answer 1

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