We use Law(X) to denote the distribution of random variable X. When ν is a probability distribution for over set Ω and Ais a subset of Ω, we use ν(A) to denote the probability that the random variable X belongs to A, when X is sampled from distribution ν. Similarly, the marginal distribution on the next N random variables for fγ,r#ν is fγ,r#ν2. We thus proved equation (22) and Lemma 3 is proved. Lemma 4 Suppose Z1( |A) and Z2( |A) are two conditional distribution with range RN, and for all values of a Ω, Wp(Law(Z1(a)),Law(Z2(a))) c (23) where Z1(a) Z1( |A= a), and Z2(a) Z2( |A= a).

Computational Thinking (CT) is a foundational problem-solving skill, and gamified programming environments are a widely adopted approach to cultivating it. While large language models (LLMs) provide on-demand programming support, current applications rarely foster CT development. We present MazeMate, an LLM-powered chatbot embedded in a 3D Maze programming game, designed to deliver adaptive, context-sensitive scaffolds aligned with CT processes in maze solving and maze design. We report on the first classroom implementation with 247 undergraduates. Students rated MazeMate as moderately helpful, with higher perceived usefulness for maze solving than for maze design. Thematic analysis confirmed support for CT processes such as decomposition, abstraction, and algorithmic thinking, while also revealing limitations in supporting maze design, including mismatched suggestions and fabricated algorithmic solutions. These findings demonstrate the potential of LLM-based scaffolding to support CT and underscore directions for design refinement to enhance MazeMate usability in authentic classrooms.

The Hierarchical Reasoning Model (HRM) has impressive reasoning abilities given its small size, but has only been applied to supervised, static, fully-observable problems. One of HRM's strengths is its ability to adapt its computational effort to the difficulty of the problem. However, in its current form it cannot integrate and reuse computation from previous time-steps if the problem is dynamic, uncertain or partially observable, or be applied where the correct action is undefined, characteristics of many real-world problems. This paper presents HRM-Agent, a variant of HRM trained using only reinforcement learning. We show that HRM can learn to navigate to goals in dynamic and uncertain maze environments. Recent work suggests that HRM's reasoning abilities stem from its recurrent inference process. We explore the dynamics of the recurrent inference process and find evidence that it is successfully reusing computation from earlier environment time-steps.