Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 4

Probability Level 2

Alice and Bob are playing a game. At the beginning of the game, there is a non-integer rational number written on the blackboard, which we denote as the starting number . The two players take turns one by one. In each turn, the player chooses any non-zero integer $n$ , and replaces the existing number $a$ on the blackboard by $(a+1/n)$ . Alice takes the first turn. Bob wins if, at any moment, the number on the blackboard is an integer. Alice wins otherwise. For which values of the starting number does Alice have a winning strategy?

$(a)$ $\dfrac{2}{3}$

$(b)$ $\dfrac{1}{5}$

$(c)$ $\dfrac{3}{7}$

$(d)$ $\dfrac{2}{5}$

This problem is a part of Tessellate S.T.E.M.S.

b, c,a a, b All of these c, d, b a, c d, b, a

2 solutions

Srijan Banerjee
Jan 10, 2018

Note that atleast for some 'n' , the gcd of (pn+1, qn) has to be 1 where the fraction is p/q , as Alice starts the game first, obviously she won't choose that 'n' where 1/n adds to p/q, such that their gcd isn't = 1....so the answer is all of it

See every time bob chooses a number n... if alice chooses - n... she is done... there is no strategy required to chose the initial number... as alice will always win

Debayan Ganguly - 3 years, 4 months ago

Vicky Ricky
Jan 3, 2018

$1/5+1/n=(n+5)/(5n)$ and now see whether $gcd(n+5,5n)$ can be 1 similarly for other options now u will see that there is a chance for the $gcd$ to become 1 for all cases for some integer n .Thus Alice has winning strategy as if $gcd$ become 1 then it's a Rational no not an integer .so all of these.

Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 4

2 solutions

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