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3475One Ternary and Two Binary Assays With Three Bad Modes Per ComponentYou may run one ternary assay and two binary assays non-adaptively. Either nothing is wrong, or exactly one component is wrong and it can be in one of three bad modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅5667Hat-Check Derangement CountSix guests check distinct hats. The attendant returns the hats in a random order. In how many of the possible return orderings does no guest receive their own hat?脑筋急转弯中等数值题未尝试免费5669Five Gifts Four Children SurjectionFive distinct gifts are distributed to 4 distinct children so that every child receives at least one gift. In how many ways can this be done?脑筋急转弯中等数值题未尝试免费5670Coprime-to-2357 Count up to 1000How many integers from 1 to 1000 are divisible by none of 2, 3, 5, or 7?脑筋急转弯中等数值题未尝试免费5671At-Least-One Fixed Point CountOf the 120 permutations of the labels 1,2,3,4,5 , how many fix at least one label in its original position?脑筋急转弯中等数值题未尝试免费5672Trilingual Survey None-CountIn a group of 100 people, 50 speak French, 40 speak German, 35 speak Spanish; 18 speak both French and German, 15 both French and Spanish, 12 both German and Spanish, and 6 speak all three. How many speak none of these three languages?脑筋急转弯简单数值题未尝试免费5673Totient of 60 by SieveHow many integers in the range 1 to 60 are relatively prime to 60 (share no common factor greater than 1 with 60)?脑筋急转弯中等数值题未尝试免费5675Seating with Forbidden Own-ChairFour people, each with an assigned chair, are reseated so that no person sits in their own assigned chair, and additionally person 1 must not sit in chair 2. How many seatings are valid?脑筋急转弯困难数值题未尝试免费5676Seven Tasks Four Machines All BusySeven distinct tasks are each assigned to one of 4 distinct machines. In how many assignments is every machine used at least once?脑筋急转弯中等数值题未尝试免费5699Sorting Five With Fewest ComparisonsYou must sort 5 distinct numbers using only pairwise comparisons, each of which returns which of the two compared elements is larger. What is the information-theoretic lower bound on the worst-case number of comparisons any comparison-based sorting algorithm needs, AND is that bound actually achievable for 5 elements? Give the minimum worst-case number of comparisons that guarantees a full sort.脑筋急转弯中等brainteaser未尝试免费5700Ten Hats in a Line10 players stand in a line. Each wears a red or blue hat, assigned independently by a fair coin. Each player sees all hats in FRONT of them but not their own nor those behind. Starting from the back of the line, each player in turn announces a single guess of their own hat color, heard by everyone. They agree on a strategy beforehand (no communication after hats are placed except the public guesses). Using the optimal parity strategy, how many of the 10 players are GUARANTEED to guess correctly regardless of the hat assignment?脑筋急转弯中等brainteaser未尝试免费5701Guess the Number 1 to 1000An adversary picks a secret integer between 1 and 1000 inclusive. You may ask yes/no questions, each answered truthfully, and you may choose each question adaptively based on previous answers. What is the minimum number of questions that guarantees you can determine the secret number in the worst case?脑筋急转弯简单brainteaser未尝试免费5702One Poisoned Bottle, Binary TestersYou have 1000 bottles of wine, exactly one of which is poisoned. A tester who drinks any amount containing the poison dies after exactly the same fixed delay, and you can have each tester sip from any combination of bottles simultaneously in a single round (results observed after the delay, before the celebration). If you only get ONE round of testing, what is the minimum number of testers needed to guarantee identifying the poisoned bottle?脑筋急转弯中等brainteaser未尝试免费5703Eight Coins, One Known LighterYou have 8 visually identical coins; exactly one is counterfeit and is known to be LIGHTER than the rest. Using a two-pan balance scale (left-heavy / right-heavy / balanced per weighing), what is the minimum number of weighings that guarantees identifying the light coin in the worst case? Weighings may be adaptive.脑筋急转弯简单brainteaser未尝试免费5704Heaviest and Runner-UpYou have 8 coins of pairwise-distinct weights and a balance scale that compares two single coins and tells you which is heavier. What is the minimum number of pairwise weighings, in the worst case, needed to identify BOTH the heaviest coin and the second-heaviest coin? (This is the classic tournament problem.)脑筋急转弯中等brainteaser未尝试免费5706How Many Coins in Three Weighings (Known Heavy)Among a pile of identical-looking coins exactly one is counterfeit and is known to be HEAVIER than the others. With a two-pan balance scale (each weighing returns left-heavy, right-heavy, or balanced) and exactly 3 weighings allowed, what is the LARGEST number of coins for which you can always guarantee identifying the heavy one? Weighings may be adaptive.脑筋急转弯简单brainteaser未尝试免费5707100 Prisoners and 100 Boxes100 prisoners are numbered 1 to 100. In a room, 100 boxes each contain one slip with a distinct number 1 to 100, placed by a uniformly random permutation. Each prisoner enters alone, may open at most 50 boxes, must find the slip bearing his own number, then leaves without communicating or altering anything. All 100 must succeed for the group to win. Using the optimal strategy (each prisoner opens the box with his number, then the box whose number matches the slip just found, following the permutation cycle), the win probability equals 1 minus the sum of 1/k for k from 51 to 100. To the nearest whole percent, what is this winning probability?脑筋急转弯困难brainteaser未尝试面试订阅5708Reference Coin Boosts CapacityExactly one coin among a pile is counterfeit, with weight different from the genuine ones, but you do NOT know whether it is heavier or lighter. You also have ONE extra coin that is guaranteed genuine, which you may place on the scale freely. Using a two-pan balance (each weighing returns left-heavy, right-heavy, or balanced) with exactly 3 weighings, what is the LARGEST number of suspect coins for which you can always both identify the fake and determine its direction? Weighings may be adaptive.脑筋急转弯困难brainteaser未尝试面试订阅5709Two Light Fakes Among SixYou have 6 visually identical coins. Exactly TWO of them are counterfeit and each counterfeit weighs the same known-lighter amount (both fakes are equally light); the other four are genuine and equal. Using a two-pan balance scale (each weighing returns left-heavy, right-heavy, or balanced), what is the minimum number of weighings that guarantees identifying WHICH two coins are the light pair in the worst case? Weighings may be adaptive.脑筋急转弯困难brainteaser未尝试面试订阅5710Digital Scale, Subset WeighingYou have 8 coins; exactly one is counterfeit and is KNOWN to be lighter than each genuine coin (all genuine coins weigh the same). Instead of a balance, you have a DIGITAL scale that reports the exact total weight of any subset of coins you place on it. Each placement-and-reading counts as one weighing. What is the minimum number of weighings that always identifies the light coin, and can a single weighing ever suffice? Give the minimum number of weighings.脑筋急转弯简单brainteaser未尝试免费