To most, Lewis Carroll is best known 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 contributions was a book of mathematical puzzles that he called “Pillow Problems.” They are so named because Carroll devised them in bed to distract himself from anxious thoughts while falling asleep. He wrote that while stirring in bed, he had two choices: “either to submit to the fruitless self-torture of going through some worrying topic, over and over again, or else to dictate to myself some topic sufficiently absorbing to keep the worry at bay. A mathematical problem is, for me, such a topic…” I personally relate to Carroll’s situation. Most nights of my life, I fall asleep while mulling over 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 bottom of today’s article. Be careful not to read too far ahead if you’re still working on that puzzle!
Puzzle #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 which color it is. You take a white marble from your pocket and add it to the bag. Then you shake up the two marbles in the bag, reach in, and pull a random one out. It happens to be white. What are the chances that the other marble in the bag is also white?
Don’t be deceived by the simple setup. This puzzle is famous for defying people’s intuitions. If you struggle to crack it, think it over while falling asleep tonight. It might at least quell your worries.
We will post the solution next Monday along with a new puzzle. Do you know a great puzzle that you think we should cover here? Send it to us: firstname.lastname@example.org
You can find this week’s solution and the following week’s puzzle here.
Solution to Puzzle #3: Calendar Cubes
Last week’s puzzle asked you to design a functioning pair of calendar cubes. Remember, a cube only has six faces. Every month has an 11th and a 22nd day, so the digits 1 and 2 must appear on both cubes, or else these days couldn’t be rendered. Notice that both cubes also need a 0. This is because the numbers 01, 02, …, and 09 all need representation, and if only one cube had a 0, there wouldn’t be enough faces on the other cube to house all nine of the other digits. This leaves us with three unoccupied faces on each cube, for a total of six more spots. However, there are seven digits remaining that need a home (3, 4, 5, 6, 7, 8, and 9). How can we squeeze seven digits onto six faces? The trick is that a 9 is an inverted 6! Beyond that realization, several assignments 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, flip that 6 upside down and, by the skin of our teeth, we have every date covered.
There’s an economy to this solution that I find beautiful. Two cubes lack the space for the task, and yet we squeak by, exploiting a quirky symmetry in our digits. Some might find this gimmicky, but this is really how store-bought calendar cubes work. If even one month of the year were extended to have 33 days, then the calendar cube market would go belly-up.
There are two natural extensions of the calendar cube puzzle to other date information. Amazingly, this theme of hair’s breadth efficiency persists across them. What if we want to add a cube that represents the day of the week? Tuesday and Thursday begin with the same letter, so we need to allow two letters on a single cube face to distinguish them: ‘Tu’ and ‘Th’. Likewise with Saturday and Sunday, which we’ll represent with ‘Sa’ and ‘Su’. Monday, Wednesday, and Friday have no conflicts so ‘M’, ‘W’, and ‘F’ will do. We find ourselves in a familiar conundrum. We have seven symbols to stuff onto only six faces of a cube. Do you see the solution? The God of Symmetry graces us again, letting ‘M’ represent Monday and, upside down, Wednesday.
We’re left with months, which I posed to you as an extra challenge last week. Can we exhibit all three-letter month abbreviations: ‘jan’, ‘feb’, ‘mar’, ‘apr’, ‘may’, ‘jun’, ‘jul’, ‘aug’, ‘sep’, ‘oct’, ‘nov’, and ‘dec’, with three more cubes containing lowercase letters? There are 19 letters that participate in some month abbreviation: ‘j’, ‘a’, ‘n’, ‘f’, ‘e’, ‘b’, ‘m’, ‘r’, ‘p’, ‘y’, ‘u’, ‘l’, ‘g’, ‘s’, ‘o’, ‘c’, ‘t’, ‘v’, ‘d’, yet again precisely one too many for the 18 faces on three cubes. Would you believe me if I told you that there is just enough symmetry in our alphabet to shoehorn every month into three cubes? The method requires that we recognize ‘u’ and ‘n’ as inversions of each other as well as ‘d’ and ‘p’. One version is depicted 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 numbering and lettering systems perfectly permit the construction of calendar cubes for days, weeks, and months, leaving no wiggle room to spare.
You might wonder: if there are 19 letters for 18 slots, why doesn’t it suffice to only combine the ‘u/n’ pair or the ‘d/p’ pair? It seems that either one would save the extra slot. The rest of the article answers that question and is a tad involved, so only stay aboard if you’re curious about the answer and don’t want to work it out on your own. The reason is that if ‘d’ and ‘p’ were split up on two different faces and only ‘u’ and ‘n’ shared a face, then we wouldn’t be able to form ‘jun’, which requires ‘u’ and ‘n’ to be representable on different cubes. On the other hand, suppose that only ‘d’ and ‘p’ share a face while ‘u’ and ‘n’ do not. June’s 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’ in order 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 the ‘u/n’ symmetry as well.
Let us know how you did on this week’s challenge in the comments.