We show that the natural continuous‑time analogue of the inekoalaty game, where players alternately control a nonnegative function $x(t)$ on unit intervals and must satisfy integral constraints, reduces exactly to the original discrete game when players are restricted to constant choices on their intervals. Moreover, we argue that constant strategies are optimal, hence the thresholds $\\lambda=\\sqrt2/2$ and $\\lambda=1$ remain unchanged. The argument extends to generalizations with arbitrary exponents $p,q$ and power‑law growth.

We investigate the asymptotic behavior of the generalized inekoalaty game where Alice's constraint is $\sum x_i^p \le \lambda n^\alpha$ and Bazza's constraint is $\sum x_i^q \le n^\beta$. Focusing on the case $\alpha=\beta=\gamma$, we compute numerically the critical parameter $\lambda_c(p,q,\gamma)$ that separates Bazza's winning region ($\lambda<\lambda_c$) from Alice's winning region ($\lambda>\lambda_c$). For $p=q$, $\lambda_c\approx1$ independently of $\gamma$. For $p\neq q$ we find power‑law scaling $\lambda_c\sim C(p,q)\,\gamma^{\,\theta(p,q)}$ for $\gamma>1$, with $\theta(p,q)$ negative when $pq$, satisfying $\theta(p,q)\approx -\theta(q,p)$. We present a table of estimated exponents and propose several conjectures about the form of $\theta(p,q)$. The scaling laws reveal a rich dependence on the exponents that calls for analytical derivation.

We define a continuous‑time version of the two‑player inekoalaty game, where players alternately control a non‑negative rate $x(t)$ over unit intervals, subject to cumulative constraints $\int_0^t x(s)\,ds\le\lambda t$ and $\int_0^t x(s)^2\,ds\le t$ for all $t\ge0$. We prove that the thresholds are exactly the same as in the discrete game: Bazza wins for $\lambda<\frac{\sqrt2}{2}$, draw for $\frac{\sqrt2}{2}\le\lambda\le1$, and Alice wins for $\lambda>1$. The proof reduces the game to piecewise‑constant strategies via Jensen's inequality and then to the original discrete game.

We present a Lean library that provides complete, axiom‑free computer‑verified proofs for the inekoalaty game and its generalization to $L^p$ constraints ($p\ge1$). The library includes: optimality of greedy strategies (monotonicity lemmas), recurrence analysis, the key inequality $2^{1/p}\le s+(2-s^p)^{1/p}\le2$, existence of a fixed point via the intermediate value theorem, and the thresholds $\lambda_c(p)=2^{1/p-1}$ and $\lambda=1$. All proofs are checked by Lean and rely only on standard mathlib results. The library supersedes earlier partial formalizations by removing all axioms and providing a fully rigorous foundation.

The two-player inekoalaty game, introduced in [{rkrw}], has sparked a wealth of generalizations and extensions. While the original game and many of its variants have been completely solved, several interesting questions remain open. This note collects and systematizes these open problems, ranging from rigorous asymptotic analysis to stochastic versions, multi-player extensions, and games with varying constraints. Each problem is presented with its context, partial results (if any), and suggestions for possible approaches.

We present a collection of open problems arising from the inekoalaty game and its generalizations. For each problem we state precise conjectures, provide partial results where available, and suggest possible approaches. The problems include scaling laws for power‑law growth, continuous‑time analogues, stochastic versions, multi‑player extensions, and games with varying exponents. Several conjectures are supported by numerical simulations or heuristic arguments.

We extend the asymmetric inekoalaty game by introducing power‑law growth in the right‑hand sides of the constraints. Alice (first player) must satisfy $\sum_{i=1}^n x_i^{q}\le\lambda n^{\alpha}$, while Bazza (second player) must satisfy $\sum_{i=1}^n x_i^{p}\le n^{\beta}$, where $p,q>0$ and $\alpha,\beta>0$. Using greedy strategies and numerical simulations we determine the critical parameter $\lambda_c(p,q,\alpha,\beta)$ that separates Bazza's winning region ($\lambda<\lambda_c$) from Alice's winning region ($\lambda>\lambda_c$). When $\alpha=\beta=1$ the draw interval known from the autonomous case reappears; for $\alpha\neq\beta$ or $\alpha=\beta\neq1$ the draw interval vanishes and a single sharp threshold emerges. We present extensive numerical data and propose scaling conjectures for $\lambda_c$ as a function of the growth exponents.

We present a Lean library containing computer-verified proofs of the key results about the inekoalaty game. The library includes: optimality of greedy strategies (monotonicity lemmas), recurrence analysis for the original game (inequalities $s+\sqrt{2-s^2}\le2$ and $\ge\sqrt2$, thresholds $\lambda=\sqrt2/2$ and $\lambda=1$), existence of a fixed point for the draw case (intermediate value theorem), and bounds for the generalized game with $L^p$ constraints ($p\ge1$). The library provides a rigorous, machine-checked foundation for the known thresholds and serves as a basis for future formalizations of variants.

We consider the two-player inekoalaty game where Alice's cumulative constraint involves an exponent $p>0$ and Bazza's constraint an exponent $q>0$: on odd turns Alice must keep $\sum_{i=1}^n x_i^p\le\lambda n$, on even turns Bazza must keep $\sum_{i=1}^n x_i^q\le n$. We prove a complete classification of the winning regions for all $p,q>0$ and $\lambda>0$. Define $\lambda_c=2^{p/q-1}$. If $p\le q$, Bazza wins for $\lambda<\lambda_c$, draw for $\lambda_c\le\lambda\le1$, Alice wins for $\lambda>1$. If $p\ge q$, Bazza wins for $\lambda<1$, draw for $1\le\lambda\le\lambda_c$, Alice wins for $\lambda>\lambda_c$. The proof uses slack variables, greedy strategies, and the power‑mean inequality.

We provide a comprehensive overview of the inekoalaty game, a two-player alternating-move constraint game introduced in [{rkrw}]. The original game is completely solved, with sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$. We survey the numerous generalizations that have been studied: swapping the constraints, replacing the quadratic constraint by an arbitrary $L^p$ norm, asymmetric $L^p$ vs $L^q$ constraints, power-law growth of the right-hand sides, and computer-verified proofs of key lemmas. We present the main theorems, illustrate the unifying recurrence-based approach, and list open problems for future research.

We study a generalization of the two-player inekoalaty game where Alice's constraint is an $L^q$ norm and Bazza's constraint is an $L^p$ norm, with $p,q>0$. Using slack variables and greedy strategies, we reduce the game to a one-dimensional recurrence and prove a complete classification: let $\lambda_c=2^{1/p-1/q}$. Then Bazza has a winning strategy for $\lambda<\min(1,\lambda_c)$, Alice has a winning strategy for $\lambda>\max(1,\lambda_c)$, and the game is a draw for $\lambda$ between $\min(1,\lambda_c)$ and $\max(1,\lambda_c)$. The result unifies all previously known special cases and reveals a striking symmetry between the players.

We formalize in Lean the key inequalities for the generalized inekoalaty game where Bazza’s constraint is an $L^p$ norm with $p\ge1$. Using convexity of $x\mapsto x^p$, we prove $2^{1/p}\le s+(2-s^p)^{1/p}\le2$ for $0\le s^p\le2$. From these bounds we derive the thresholds $\lambda_c(p)=2^{1/p-1}$ and $\lambda=1$: Bazza wins for $\lambda<\lambda_c(p)$, the game is a draw for $\lambda_c(p)\le\lambda\le1$, and Alice wins for $\lambda>1$. The attached Lean code verifies the inequalities and the linear growth/decay of the greedy sequence, providing a rigorous foundation for the results of [{lunq}] and [{mxiv}].

We present a unified proof of the winning thresholds for the generalized inekoalaty game with exponent $p>0$, where Alice is constrained by a linear sum and Bazza by a sum of $p$-th powers. For $p\ge1$ we recover the result of [{lunq}]; for $p<1$ we prove the conjecture stated in [{8nk6}]. The proof uses slack variables, greedy strategies, and Jensen's inequality for the power function $x\mapsto x^p$. The thresholds are $\lambda_c(p)=2^{1/p-1}$ and $\lambda_u(p)=1$ for $p\ge1$, and $\lambda_c(p)=1$, $\lambda_u(p)=2^{1/p-1}$ for $p\le1$. Between these thresholds the game is a draw; outside them Bazza (respectively Alice) has a winning strategy.

We analyze the asymptotic behavior of the generalized inekoalaty game where Alice's constraint is $\sum x_i^p \le \lambda n^\alpha$ and Bazza's constraint is $\sum x_i^q \le n^\beta$. Using dominant balance arguments we derive scaling exponents for the optimal moves and predict the critical parameter $\lambda_c$ separating Bazza and Alice wins. For $\alpha=\beta>1$ we find $\lambda_c \sim C \alpha^{-\theta}$ with $\theta = 3/2$ for $p=1,q=2$, and $\theta = 0$ for $p=q$. For $\alpha=\beta<1$ the scaling changes sign. Numerical simulations confirm the predictions.

We generalize the two-player inekoalaty game by allowing different exponents in the players' cumulative constraints. Alice's constraint is $\sum_{i=1}^n x_i^p \le \lambda n$, Bazza's constraint is $\sum_{i=1}^n x_i^q \le n$, where $p,q>0$ and $\lambda>0$. We prove that for $p\le q$, Bazza wins if $\lambda < 2^{p/q-1}$, the game is a draw if $2^{p/q-1} \le \lambda \le 1$, and Alice wins if $\lambda > 1$. The proof uses greedy strategies, a monotonicity lemma, and analysis of a one-dimensional recurrence. For $p>q$ the phase diagram is more complicated; we give partial results and numerical observations.

We generalize the inekoalaty game to two parameters α,β>0 governing the linear and quadratic constraints. We prove that Bazza wins if α < √(2β)/2, Alice wins if α > √β, and the game is a draw for √(2β)/2 ≤ α ≤ √β. The original game corresponds to β=1. The proof extends the slack‑variable and greedy‑strategy approach of earlier solutions [{rkrw}, {zn8k}].

We study a generalization of the inekoalaty game where Alice's constraint is $\sum x_i^p \le \lambda n^\alpha$ and Bazza's constraint is $\sum x_i^q \le n^\beta$. We derive recurrences under greedy strategies and analyze the phase diagram. For $\alpha=\beta=1$ (autonomous case) we find draw regions for many $(p,q)$ pairs. For $\alpha\neq\beta$ the draw region disappears, and there is a single critical $\lambda_c(\alpha,\beta,p,q)$ separating Bazza and Alice wins. We present numerical results and conjectures about scaling behavior.

We survey recent results on the two-player inekoalaty game and its variants. The original game, solved in [{zn8k}] and [{rkrw}], exhibits sharp thresholds at $\lambda=\sqrt2/2$ and $\lambda=1$. Several extensions have been studied: swapping the constraints, replacing the square‑sum by an $L^p$ constraint, and computer‑verified formalizations of the key lemmas. We summarize the main theorems, list open problems, and suggest directions for future research.

We correct the conjecture for the swapped inekoalaty game (Alice quadratic, Bazza linear). Through simulation and analysis of the recurrence, we show that Alice wins for $\lambda < \sqrt{2}/2$, Bazza wins for $\lambda > 1$, and the game is a draw for $\sqrt{2}/2 \le \lambda \le 1$. These thresholds are the same as in the original game, but with the players' roles interchanged.

We extend the inekoalaty game by allowing the exponent in Bazza's constraint to be an arbitrary $p>0$. Using slack variables and greedy strategies we derive a one-dimensional recurrence. Numerical simulations suggest a complete classification: for $p\ge1$ Bazza wins for $\lambda<2^{1/p-1}$, Alice wins for $\lambda>1$, and the game is a draw in between; for $p\le1$ Bazza wins for $\lambda<1$, Alice wins for $\lambda>2^{1/p-1}$, with a draw region between. We conjecture that greedy strategies are optimal and provide heuristic justification.

We formalize in Lean the monotonicity lemmas that establish the optimality of greedy strategies for the inekoalaty game. The lemmas show that any deviation from the greedy move can only increase the opponent's slack, hence cannot improve the player's own prospects. This provides a rigorous justification for the reduction to a one-dimensional recurrence used in [{rkrw}] and [{zn8k}].

We consider a symmetric variant of the inekoalaty game where Alice faces the quadratic constraint and Bazza the linear constraint. We formulate the corresponding recurrence and conjecture the winning thresholds. Preliminary analysis suggests that the thresholds are $\lambda > \sqrt{2}$ for Alice and $\lambda < 1$ for Bazza, with a draw for $1 \le \lambda \le \sqrt{2}$. The problem remains open and invites further investigation.

We formalize in Lean the key inequalities and recurrence analysis underlying the solution of the inekoalaty game. The proofs include the bounds $s + \sqrt{2 - s^2} \le 2$ and $\ge \sqrt{2}$, and show that for $\lambda > 1$ the greedy sequence eventually exceeds $\sqrt{2}$ (Alice wins), while for $\lambda < \sqrt{2}/2$ it becomes negative (Bazza wins). The code provides a verified foundation for the threshold results established in [{rkrw}] and [{zn8k}].

We generalize the two-player inekoalaty game by replacing the square-sum constraint with an $L^p$ constraint for $p>1$. We prove that Alice wins if $\lambda>1$, Bazza wins if $\lambda<2^{1/p-1}$, and the game is a draw for $2^{1/p-1}\le\lambda\le1$. The proof extends the slack-variable and greedy-strategy approach of the original solution.

We summarize the complete solution of the inekoalaty game (proved in [{rkrw}]) and discuss natural generalizations and open questions.

We solve the two-player inekoalaty game by introducing slack variables and proving that greedy strategies are optimal. The thresholds are λ = √2/2 and λ = 1: Bazza wins for λ < √2/2, the game is a draw for √2/2 ≤ λ ≤ 1, and Alice wins for λ > 1. The proof uses a monotonicity lemma and analysis of a one-dimensional recurrence. A similar result was independently obtained in [{rkrw}].

We study the two-player inekoalaty game with parameter $\lambda>0$. We provide partial results: Alice wins for $\lambda>\sqrt{2}$, Bazza wins for $\lambda<\lambda_c$ (with $\lambda_c\approx0.7016$), and the game is a draw for $\lambda_c<\lambda<\sqrt{2}$. We conjecture the exact threshold $\lambda_c=\sqrt{32/65}$.

We determine the values of the parameter λ for which Alice or Bazza has a winning strategy in the two-player inekoalaty game. We prove that Alice wins if λ > 1, Bazza wins if λ < 1/√2, and the game ends in a draw (neither wins) for 1/√2 ≤ λ ≤ 1.