Problem 5

Alice and Bazza are playing the inekoalaty game, a two-player game whose rules depend on a positive real number

\lambda

which is known to both players. On the

n

th turn of the game (starting with

n=1

) the following happens: if

n

is odd, Alice chooses a nonnegative real number

x_n

such that

x_1+x_2+\cdots+x_n\le\lambda n

; if

n

is even, Bazza chooses a nonnegative real number

x_n

such that

x_1^2+x_2^2+\cdots+x_n^2\le n

. If a player cannot choose a suitable number

x_n

, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players. Determine all values of

\lambda

for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Step 4 of 4: Conclusion: 1/√2 is the exact threshold

Previous step Next step

\text{Alice wins}\iff\lambda>\tfrac1{\sqrt2},\qquad \text{Bazza wins}\iff\lambda<\tfrac1{\sqrt2}

Detailed analysis

Steps 1 and 3 show Alice has a winning strategy exactly when $\lambda>1/\sqrt2$ and Bazza has one exactly when $\lambda<1/\sqrt2$ . At $\lambda=1/\sqrt2$ itself, both bounding computations become equalities in the limit rather than strict inequalities, so neither player can force the other to run out of legal moves; this boundary value belongs to neither list.