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548Commute Time Between Endpoints of a PathConsider the path graph P n with vertices \ 0, 1, \ldots, n-1\ and n-1 edges (each of unit resistance), where the walk at interior vertices moves left or right with equal probability, and at the endpoints moves to the unique neighbor. (a) Compute the effective resistance R eff (0, n-1) between the two endpoints. (b) Using C(u,v) = 2m R eff (u,v), find the commute time between the two endpoints. (c) Verify for n = 4 by computing h(0 3) and h(3 0) directly.概率中等derivation未尝试免费550Expected Cover Time of the Cycle C₆A simple random walk moves on the cycle graph C 6 (vertices 0, 1, \ldots, 5). At each step, the walker moves clockwise or counterclockwise with equal probability. Starting at vertex 0, what is the expected number of steps to visit all 6 vertices (the expected cover time)?概率困难derivation未尝试面试订阅631Terminal-Variable Projection 1Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = 1 X 1+X 2+X 3 >= 2 and M n = E[Y | F n]. Is (M n) a martingale?概率简单derivation未尝试免费634Terminal-Variable Projection 4Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = X 1+X 2+X 3+X 4 and M n = E[Y | F n]. Is (M n) a martingale?概率中等derivation未尝试免费636Normalized Product Process 1Let Y 1, Y 2, ... be iid positive random variables taking values [1, 3] with probabilities ['1/2', '1/2'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(2) n. Is (M n) a martingale?概率简单derivation未尝试免费677Two-Step Recurrence Repeat 2A deterministic recurrence over mod 13 is fully determined by the ordered pair (x t, x (t+1)). How many consecutive ordered pairs force a repeated state pair?脑筋急转弯简单brainteaser未尝试免费679Queue-Regime Repeat 4A simplified venue state is (bid queue mod 6, ask queue mod 6, regime among 4 labels). How many observations force a repeated full state?脑筋急转弯中等brainteaser未尝试免费681Length-and-Sum Congruence Window 1Given 13 integers a 1, ..., a 13, show that some nonempty consecutive block has length divisible by 3 and sum divisible by 4.脑筋急转弯简单derivation未尝试免费682Sector-Liquidity Tag Collision 2Each trade is tagged by one of 4 sectors and one of 4 liquidity buckets. Show that among 17 trades, two trades must share both tags.脑筋急转弯中等brainteaser未尝试免费687Even-Length Inventory Window 2Given 10 integers x 1, ..., x 10, show that some nonempty consecutive block of even length has sum divisible by 5.脑筋急转弯简单derivation未尝试免费691Parity-Aligned Repeated 2-Block 6Show that every binary string of length 10 contains two identical consecutive 2-bit blocks whose starting positions have the same parity.脑筋急转弯简单derivation未尝试免费696Consecutive Block With Sum Minus Length Divisible by 9Given 9 integers a 1, ..., a 9, prove that some nonempty consecutive block has sum congruent to its length modulo 9.脑筋急转弯简单derivation未尝试免费698Subset Sum Divisible by 8 3Given 8 integers, prove that some nonempty subset has sum divisible by 8.脑筋急转弯中等brainteaser未尝试免费699Semicircle Load 4Twelve points lie on a circle. Show that some closed semicircle contains at least 6 of them.脑筋急转弯中等brainteaser未尝试面试订阅750Domino Strip Tiling 5How many domino tilings does a 2-by-8 board have?脑筋急转弯困难数值题未尝试面试订阅755Market-State Logic Grid 5Three analysts A, B, and C are each either truthful (always tell the truth) or lying (always make false statements). The hidden market state is either bull or bear. A says: 'The market is bull.' B says: 'Exactly one of us is truthful.' C says: 'B and C are the same type.' Determine the market state and the type of each analyst.脑筋急转弯困难brainteaser未尝试免费760Market-State Logic Grid 10Three analysts A, B, and C are each either truthful (always tell the truth) or lying (always make false statements). The hidden market state is either bull or bear. A says: 'The market is bull.' B says: 'Exactly two of us are truthful.' C says: 'At least one of B and C is lying.' Determine the market state and the type of each analyst.脑筋急转弯困难brainteaser未尝试面试订阅762Market-State Logic Grid 12Three analysts A, B, and C are each either truthful (always tell the truth) or lying (always make false statements). The hidden market state is either bull or bear. A says: 'The market is bull.' B says: 'B and C are the same type.' C says: 'A and B are both truthful.' Determine the market state and the type of each analyst.脑筋急转弯简单brainteaser未尝试免费763Market-State Logic Grid 13Three analysts A, B, and C are each either truthful (always tell the truth) or lying (always make false statements). The hidden market state is either bull or bear. A says: 'The market is bull.' B says: 'A and C are different types.' C says: 'At least one of B and C is lying.' Determine the market state and the type of each analyst.脑筋急转弯中等brainteaser未尝试免费764Market-State Logic Grid 14Three analysts A, B, and C are each either truthful (always tell the truth) or lying (always make false statements). The hidden market state is either bull or bear. A says: 'The market is bull.' B says: 'A and C are different types.' C says: 'A and B are both truthful.' Determine the market state and the type of each analyst.脑筋急转弯困难brainteaser未尝试免费