There is a famed trouble in math called The Secretary Problem . You are hiring for a line of work at your ship’s company , and you will question n people , one at a sentence . Through the interviews , you are able to rank each candidate in lodge relative to the other candidate you ’ve seen so far ( meaning if you ’ve already met with five citizenry , then you know which one was the good of the five , which was second best , and so on ) . The trouble is , after each interview , you must decide on the spot whether you want to hire that prospect or to reject them and continue the process , chance never meet someone that dispose again . What is the optimal scheme to maximize your chance of hiring the best applicant ?
The problem is famous for at least two reasonableness . One is that the optimum scheme pay you impressively good probability of discover the dependable prospect . The other is that mathematician ’ favored little constant bring in a surprise coming into court in the solution : e.
Euler ’s number , e , is about 2.7182 and is renowned for cropping up everywhere in seemingly disparate area of mathematics . You might have encountered e in tophus stratum , or if you ’ve savour compound interest on your investment , or ever looked at a bell bender , or suffer bacteria maturation , or had shock absorber on your wheel / cable car , or let yourcoffee cool . As numerical constants go , pi savour celebrity status , with its own vacation and contests for memorizing its digits . Meanwhile , e is the modest workhorse of the physical domain , dutifully holding everything together in the background , too self-respectful for limelight .
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Here ’s the solution to The Secretary Problem : Always refuse the first 1 / e fraction of candidates out of paw ( the first ~37 % of applier ) . After that , hire the first candidate you meet who is better than every other one you ’ve met so far ( if you never adjoin such a candidate , tough lot ) . Amazingly , this simple scheme devote you a roughly 37 % ( again , 1 / tocopherol ) chance of get hold the best campaigner , disregardless of how many applicant there are . Even with million of applicant , you have a better than one in three chance of discover the top one among them!Psychological researchsuggests that when people are faced with real - life secretary problems , they tend to curtail their search prematurely , leading to suboptimal outcomes . So next sentence you ’re hound for the cheapest gas on the highway or deciding whether toapply for an apartment vs. go on your hunting , view put on the secretary problem approach and seeking for a little longer than you might normally be incline .
There is a whole robust hypothesis focus solely on stopping rules , i.e. , when to stop a process to achieve a desired end . This week ’s Gizmodo Monday Puzzle does n’t imply Euler ’s number or sophisticated math of any kind , but it does call for you when to cease .
Did you miss last week ’s puzzle ? Check it outhere , and observe its solution at the bottom of today ’s clause . Be careful not to register too far out front if you have n’t solve last hebdomad ’s yet !
Puzzle #16: Turning Red
You shuffle a normal deck of card face down and then begin to flip over cards from the top of the pack of cards , one at a time , place them confront up on a table . At any sentence ( but only once ) , you’re able to choose to finish , and if the next identity card is scarlet , then you win . If you never discontinue , then by default you ’re squeeze to choose the last card ( again , you gain if it ’s ruddy ) . Is there a strategy that maximizes your chance of get ahead this game ? If so , what is it ? If not , why not ? You must shuffle the card thoroughly and are not allowed to chisel in any way ( like by marking cards ) . You may only discover the board that you flip and take when to intercept .
Scroll down for the solution .
Solution to Puzzle #15: Spell It Out
Last week , I gave you a novel way tolook at numbers . Let ’s take them one by one .
What is the smallest number that comprise the letter “ a ” when spell out ? response : one thousand . Considering “ a ” is one of the most usual letters in the alphabet , it ’s surprising how rare it is in our mathematical figure . The little number that contains a “ c ” is one octillion .
There is only one phone number that , when spell out , has its letter in alphabetic edict . What is it ? Answer : forty .
There is also only one number with its missive in reverse alphabetical lodge . What is it ? solvent : one . I could n’t fend lift the reply into the question .
Imagine we fill a dictionary with the first trillion number in alphabetical order . What is the first odd number in the dictionary ? resolution : eight billion eighteen million eighteen thousand eight hundred eighty five , or 8,018,018,885 . For direct contrast , the first even bit in the dictionary is 8 . you’re able to see the first several entrieshere .
Solution to Puzzle #16: Turning Red
Many citizenry have a unassailable intuition that they can reach an edge in this secret plan . A common idea is to stop as shortly as there are more red cards remain in the deck than opprobrious cards . The surprising winding is that there is no strategy that give you undecomposed than a 50/50 chance of stopping on a scarlet card . In fact , no strategy pay you a bad than 50/50 luck either . ready up any haywire scheme you care , and it will have no effect .
A slick way to see this is to consider the following true pointless plot . We ’ll have the same setup : a shuffled deck of cards , thumb over one posting at a time , and arrest whenever you please , except this prison term when you stop , you wait at the bottom card of the pack of cards alternatively of the top . If it ’s flushed , you make headway . The bottom card never changes and is secure as either red or black from the start , so clearly any strategy to beat a 50/50 chance in this game is doomed . The key observation is that the probability in our original game are indistinguishable at every step to the probabilities in this pathetic variant biz . blockade flip-flop at any breaker point — is it any more probable that the top notice in the deck is violent than the bottom batting order ? possibly at certain times there is a greater than 50 % opportunity that the top card is red , but at such multiplication there is also an equivalent prospect that the bottom batting order is flushed , or any of the remain card for that topic . So regardless of when you stop , you ca n’t do any better than a secret plan where you merely ruffle the cards and then glance at the bottom one , which will only be ruddy one-half of the time .
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