QR Video OA: Advanced Math and Reasoning

A six-question QR video OA gives about five minutes per question and asks candidates to explain math, probability, Markov-chain, calculus, and linear-algebra reasoning on camera. Requirements The QR video OA (also delivered through a VidCruiter-style recorder) contains six to seven short-answer questions, roughly five minutes per question, with the camera on and a spoken explanation required for each. Topics surfaced: Brain teaser: on a square grid, cells initially contain fungus; if a cell has at least two infected orthogonal neighbors, it becomes infected. Find the minimum initial infected cells needed to infect the whole board. Chain-rule differentiation. Leibniz integral rule. Standard Markov-chain reasoning. Newton method root-finding convergence. Linear-algebra question around orthogonal vectors / orthogonal matrices and related extensions. Probability on a graph: an ant starts at one vertex of a cube and at each step moves along one of the three incident edges uniformly at random; find the expected number of edges traversed before it first reaches the vertex opposite the start. Notes This round is less about polished proof and more about quickly recognizing the tool. For the fungus grid, model the infection as monotone cellular automaton growth: corner/edge placement matters because each new cell needs two infected neighbors. For Markov chains, be ready to define states, transition probabilities, absorbing states, and expected hitting time. For Newton convergence, state the iteration x_{n+1} = x_n - f(x_n)/f'(x_n) and discuss local convergence when the initial point is close and f'(root) != 0. The video format penalizes silence. Even if the final answer is incomplete, narrate definitions, assumptions, and a small example. Preparation Review chain rule, product/quotient rule, and differentiating under the integral sign. Do one-page refreshers on Markov chains: transition matrix, stationary distribution, absorption probability, and expected hitting time. Practice explaining Newton's method convergence and one failure mode, such as derivative near zero or poor initial guess. Set up the cube-ant expected-value problem by symmetry: group vertices by distance from the start (1 start, 3 adjacent, 3 at distance 2, 1 opposite) and solve the resulting linear system for expected hitting time. Rehearse 5-minute spoken solutions; this is a speed-and-communication round, not a written exam.