To most people, Lewis Carroll is best known as an eccentric author Alice’s Adventures in WonderlandBut did you know that he was also an avid puzzler and published mathematician? Among his many contributions was a book on mathematical puzzles called Pillow Problems. They are named because Carroll created them in bed to distract himself from disturbing thoughts during sleep. Stirring in bed, he wrote, he had two options: “Either submit to the fruitless self-torture of going through some troubling subject, over and over, or dictate to myself a subject interesting enough to keep anxiety at bay.” Mathematical problem He isfor me, such a subject…” I personally relate to Carol’s situation. Most nights of my life I fall asleep while thinking of a puzzle and have found it an effective antidote to a restless head.
Did you miss last week’s challenge? check it out here, and find its solution at the end of today’s article. Be careful not to read too far if you’re still working on this puzzle!
Mystery #4: Lewis Carroll’s pillow problem
You have an opaque bag containing one marble that has a 50/50 chance of being black or white, but you don’t know what color it is. You take a white marble from your pocket and add it to the bag. Then you shake the two balls into the bag, reach inside, and draw one at random. It happens to be white. What are the odds that the other marble in the bag is also white?
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Don’t be fooled by the simple setup. This puzzle is famous for challenging people’s intuition. If you’re struggling to break it, think about it while you sleep tonight. It may at least ease your concerns.
We will post the solution next Monday with a new puzzle. Know of a cool mystery that you think we should cover here? Send it to us: [email protected]
Solve puzzle number 3: calendar cubes
past weeks puzzle You are asked to design an efficient pair of calendar blocks. Remember that a cube has only six faces. Each month has an 11th and 22nd day, so the numbers 1 and 2 must appear on both cubes, otherwise these days will not be displayed. Note that both cubes also need a 0. This is because the numbers 01, 02, … and 09 all need to be represented, and if only one cube contains a 0, there won’t be enough faces on the other cube to house the nine of the other numbers. This leaves us with three empty faces on each cube, for a total of six more points. However, there are seven remaining numbers that need a home (3, 4, 5, 6, 7, 8 and 9). How can we compress seven numbers on six faces? The trick is that the number 9 is an inverted 6! Beyond this realization, many tasks operate. For example, put 3, 4, and 5 on one cube, and 6, 7, and 8 on the other cube. When the 9th day comes, turn the number 6 upside down and, together with the skin of our teeth, cover each date.
There is an economy to this solution which I find beautiful. Two cubes lack room for the task, and yet we resort to exploiting the odd symmetry in our numbers. Some may find this strange, but that’s how store-bought calendar cubes work. If one month of the year were extended to include 33 days, the market for calendar cubes would go even higher.
There are two natural extensions of the calendar cube puzzle to other date information. Amazingly, the theme of hair-widening efficiency continues across. What if we want to add a cube that represents the day of the week? Tuesday and Thursday start with the same letter, so we need to allow two letters on one face of the cube to distinguish them: “Tu” and “Th”. Likewise with Saturday and Sunday, which we will represent with “Sa” and “Su”. Monday, Wednesday, and Friday don’t have any conflicts, so “M,” “W,” and “F” will do. We find ourselves in a familiar dilemma. We have seven symbols to fill on only six sides of the cube. Do you see the solution? The god of symmetry decorates us again, making the letter “M” represent Monday, and Wednesday upside down.
We have months left, which I brought to you as an extra challenge last week. Can we display all three letter abbreviations of the month: “jan”, “feb”, “mar”, “apr”, “may”, “jun”, “jul”, “aug”, “sep”, “oct” , “nov” and “dec” with three cubes containing lowercase letters? There are 19 letters involved in the abbreviations of a month: “j,” “a,” “n,” “f,” “e,” “b,” “m,” “r,” “p,” “y,” u”, “l”, “g”, “s”, “o”, “c”, “t”, “v”, “d”, again one selection too many for the 18 faces on three cubes. Would you believe me if I told you there is Just Enough consistency in our alphabet to turn a shoe every month into three blocks? The method requires that we recognize “u” and “n” as inversions of each other as well as “d” and “p”. One copy is shown below:
cube = 1 [j, e, r, y, g, o]
cube = 2 [a, f, s, c, v, (n/u)]
Cube 3 = [b, m, l, t, (d/p), (n/u)]
Somehow, the few symmetries in our letter and numbering systems perfectly allow calendar cubes to be built for days, weeks, and months, leaving no wiggle room.
You might be wondering: if there are 19 characters for 18 digits, why isn’t it enough to combine the “u/n” pair or the “d/p” pair? Looks like either one will provide the extra slot. The rest of the article answers that question and is involved, so only stay on board if you’re curious about the answer and don’t want to work it out on your own. The reason is that if the letters ‘d’ and ‘p’ are split into two different faces and only ‘u’ and ‘n’ share one face, we cannot form ‘jun’, which requires ‘u’ and ‘n’ to be represented as different cubes. On the other hand, suppose only ‘d’ and ‘p’ share a face while ‘u’ and ‘n’ do not. The June abbreviation insists that “j”, “u”, and “n” be on different cubes:
cube = 1 [j, …]
cube = 2 [u,…]
Cube 3 = [n,…]
Furthermore, ‘a’ must share a cube with ‘u’ to form ‘jan’:
cube = 1 [j, …]
cube = 2 [u, a, …]
Cube 3 = [n,…]
But then how do we make “aug”? The letters “a” and “u” share a face. The only way out is to use “u/n” symmetry as well.
Tell us how you handled this week’s challenge in the comments.
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