Gizmodo Monday Puzzle: Can You Solve Lewis Carroll’s Tricky ‘Pillow Problem’?

To most , Lewis Carroll is best know as the whimsical author of Alice ’s Adventures in Wonderland , but did you know that he was also an avid puzzler and published mathematician ? Among his many contribution was a leger of mathematical puzzle that he called “ Pillow Problems . ” They are so named because Carroll devised them in bed to disorder himself from anxious thoughts while falling at peace . He pen that while stirring in bed , he had two choices : “ either to state to the fruitless ego - torture of going through some worrying topic , over and over again , or else to order to myself some topic sufficiently absorbing to keep the worry at embayment . A mathematical job is , for me , such a issue … ” I in person relate to Carroll ’s situation . Most night of my life history , I light asleep while mulling over a puzzler and have found it an effective antidote to a restless head .

Did you miss last week ’s challenge ? ensure it outhere , and find its solution at the bottom of today ’s clause . Be careful not to read too far onward if you ’re still work on that teaser !

Puzzle #4: Lewis Carroll’s Pillow Problem

You have an unintelligible bag containing one marble that has a 50/50 chance of being black or white , but you do n’t know which gloss it is . You take a livid marble from your pocket and add it to the bag . Then you agitate up the two marble in the bag , get hold of in , and pull a random one out . It materialise to be clean . What are the chance that the other marble in the bagful is also ashen ?

Do n’t be deceived by the simple setup . This mystifier is noted for defying people ’s intuitions . If you struggle to break it , think it over while falling at peace tonight . It might at least quell your worries .

We will post the answer next Monday along with a newfangled puzzler . Do you know a corking puzzle that you think we should cover here ? Send it to us:[email protected ]

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you could rule this calendar week ’s solution and the following workweek ’s puzzlehere .

Solution to Puzzle #3: Calendar Cubes

Last week’spuzzleasked you to contrive a functioning twain of calendar cubes . Remember , a cube only has six faces . Every calendar month has an 11th and a twenty-second twenty-four hours , so the digits 1 and 2 must seem on both cubes , or else these years could n’t be render . acknowledge that both third power also need a 0 . This is because the numeral 01 , 02 , … , and 09 all necessitate representation , and if only one regular hexahedron had a 0 , there would n’t be enough faces on the other regular hexahedron to house all nine of the other digits . This leave us with three untenanted faces on each cube , for a total of six more spots . However , there are seven digits stay on that need a home base ( 3 , 4 , 5 , 6 , 7 , 8 , and 9 ) . How can we squash seven digits onto six face ? The trick is that a 9 is an inverted 6 ! Beyond that actualization , several duty assignment work . For example , put 3 , 4 , and 5 on one cube and 6 , 7 , and 8 on the other one . When the 9th rolls around , riff that 6 upside down and , by the skin of our teeth , we have every particular date overcompensate .

There ’s an economy to this resolution that I notice beautiful . Two cubes miss the distance for the job , and yet we creak by , exploiting a way-out symmetry in our finger . Some might get hold this gimmicky , but this is really how memory board - buy calendar cube shape . If even one month of the year were extended to have 33 days , then the calendar square block market place would go belly - up .

There are two born extensions of the calendar cube puzzle to other date data . Amazingly , this root of hair ’s breadth efficiency prevail across them . What if we desire to add a cube that interpret the day of the week ? Tuesday and Thursday set about with the same letter , so we require to let two varsity letter on a single block face to secernate them : ‘ Tu ’ and ‘ atomic number 90 ’ . Likewise with Saturday and Sunday , which we ’ll represent with ‘ Sa ’ and ‘ Su ’ . Monday , Wednesday , and Friday have no conflict so ‘ M ’ , ‘ W ’ , and ‘ F ’ will do . We find ourselves in a familiar conundrum . We have seven symbols to stuff onto only six case of a cube . Do you see the solvent ? The God of Symmetry adorn us again , lease ‘ chiliad ’ represent Monday and , upside down , Wednesday .

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We ’re get out with months , which I pose to you as an extra challenge last week . Can we expose all three - varsity letter calendar month abbreviations : ‘ jan ’ , ‘ feb ’ , ‘ mar ’ , ‘ apr ’ , ‘ may ’ , ‘ jun ’ , ‘ jul ’ , ‘ aug ’ , ‘ sep ’ , ‘ oct ’ , ‘ nov ’ , and ‘ dec ’ , with three more third power check lower-case letter letter ? There are 19 letter that take part in some calendar month abbreviation : ‘ j ’ , ‘ a ’ , ‘ n ’ , ‘ f ’ , ‘ Es ’ , ‘ b ’ , ‘ m ’ , ‘ r ’ , ‘ phosphorus ’ , ‘ atomic number 39 ’ , ‘ atomic number 92 ’ , ‘ l ’ , ‘ GB ’ , ‘s ’ , ‘ o ’ , ‘ c ’ , ‘ thymine ’ , ‘ v ’ , ‘ d ’ , yet again incisively one too many for the 18 faces on three cubes . Would you believe me if I tell you that there is just enough symmetry in our alphabet to shoehorn every month into three cubes ? The method acting need that we recognize ‘ u ’ and ‘ n ’ as inversions of each other as well as ‘ d ’ and ‘ phosphorus ’ . One variation is depicted below :

Cube 1 = [ j , e , r , y , g , o ]

Cube 2 = [ a , f , s , c , v , ( n / u ) ]

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Cube 3 = [ b , m , l , t , ( d / p ) , ( n / u ) ]

Somehow , the few symmetries in our numbering and inscription arrangement dead permit the construction of calendar cubes for twenty-four hour period , weeks , and calendar month , leaving no wiggle elbow room to spare .

You might wonder : if there are 19 letters for 18 slot , why does n’t it suffice to only combine the ‘ u / n ’ pair or the ‘ d / p ’ pair ? It seems that either one would economise the additional slot . The residual of the clause respond that interrogative sentence and is a shade ask , so only remain on base if you ’re rummy about the answer and do n’t desire to work it out on your own . The reason is that if ‘ d ’ and ‘ p ’ were split up on two unlike faces and only ‘ u ’ and ‘ n ’ share a face , then we would n’t be capable to form ‘ jun ’ , which need ‘ u ’ and ‘ n ’ to be representable on different cubes . On the other bridge player , suppose that only ‘ vitamin D ’ and ‘ phosphorus ’ share a cheek while ‘ atomic number 92 ’ and ‘ n ’ do not . June ’s abbreviation insist that ‘ j ’ , ‘ u ’ , and ‘ n ’ be on different cubes :

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