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651Fair Corridor Hit Probability 1A symmetric random walk starts at 2 on the integer line and stops when it first hits 0 or 7. What is the probability that it hits 7 before 0?概率简单derivation未尝试免费656Fair Corridor Exit Time 1A symmetric random walk starts at 1 and stops when it first hits 0 or 6. What is the expected stopping time?概率中等数值题未尝试免费661Lazy Walk Exit Time 1A lazy symmetric walk starts at 2 and, each period, moves +1 with probability 3/8, moves -1 with probability 3/8, and stays put with probability 1/4. It stops when it first hits 0 or 8. What is the expected stopping time?概率简单数值题未尝试免费667Biased Corridor Hit Probability 1A random walk starts at 2, moves +1 with probability 3/5 and -1 with probability 2/5, and stops when it first hits 0 or 7. What is the probability that it reaches 7 before 0?概率中等数值题未尝试免费673Scaled-Step Exit Time 3A fair random walk starts at 3 and moves by +3 or -3 with equal probability each step. It stops when it first hits -9 or 9. What is the expected stopping time?概率中等数值题未尝试免费674Scaled-Step Exit Time 2A fair random walk starts at 4 and moves by +4 or -4 with equal probability each step. It stops when it first hits 0 or 16. What is the expected stopping time?概率中等derivation未尝试免费676Inventory-Spread State Repeat 1A simulator records the pair (inventory mod 9, spread bucket among 5 buckets) once per minute. What is the minimum number of recorded minutes that guarantees two records have exactly the same pair?脑筋急转弯简单数值题未尝试免费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678Bitmask-and-Mode Repeat 3A trading engine state is described by a 4-bit flag pattern and one of 3 operating modes. What is the smallest number of snapshots that guarantees a repeated full state?脑筋急转弯中等数值题未尝试免费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680Automaton Visit Repeat 5A deterministic signal automaton has 81 possible internal states. Starting from any state, after how many visited states must some state have appeared at least twice?脑筋急转弯困难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683Venue-Session Tag Collision 3Each quote is classified by one of 3 venues and one of 6 session buckets. How many quotes guarantee that two quotes share the same ordered pair of tags?脑筋急转弯中等brainteaser未尝试免费684Parity-and-Desk Match 4On a floor with 5 desks, each ticket also has one of 2 parity labels. How many tickets guarantee that two tickets share both the same desk and the same parity label?脑筋急转弯中等brainteaser未尝试面试订阅685Calendar-Digit Collision 5A process tags each alert by one of 3 weekday classes and one of 5 last-digit classes. Show that among 16 alerts, two must share both tags.脑筋急转弯困难数值题未尝试免费686Twelve Updates in One Second 1Twelve quote updates occur in a 1-second interval. Prove that two updates are at most 1/11 second apart.脑筋急转弯简单数值题未尝试免费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688Shifted-Congruence Block 3Given 15 integers y 1, ..., y 15, show that some nonempty consecutive block has sum congruent to twice its length modulo 13.脑筋急转弯困难derivation未尝试面试订阅690Repeated State in a 20-State Engine 5A deterministic trading engine is described by a state pair (inventory mod 4, cash-balance bucket mod 5). Starting from any state and recording one state per observation, how many observations are enough to guarantee that some state is seen at least twice?脑筋急转弯中等数值题未尝试面试订阅