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 3 of 4: Bazza eventually leaves Alice with no legal move

Previous step Next step

\lambda<\tfrac1{\sqrt2}\ \Longrightarrow\ \text{Bazza wins}

Detailed analysis

Summing step 2 over the $k$ pairs $(x_1,x_2),\ldots,(x_{2k-1},x_{2k})$ gives $x_1+\cdots+x_{2k}\ge k\sqrt2$ under Bazza's strategy, regardless of Alice's play. For Alice to have a legal move $x_{2k+1}\ge0$ on turn $2k+1$ , she needs $x_1+\cdots+x_{2k}\le\lambda(2k+1)$ , hence $k\sqrt2\le\lambda(2k+1)$ . If $\lambda<1/\sqrt2$ , then as $k\to\infty$ the left side grows like $\sqrt2\,k$ while the right side grows like $2\lambda k<\sqrt2\,k$ , so this inequality fails for all sufficiently large $k$ ; at that point Alice has no legal move and loses.