To most, Lewis Carroll is best known as the whimsical author of Alice’s Adventures in Wonderland, but did you know he was also an avid puzzler and published mathematician? Among his many contributions was a book of math puzzles he called “Pillow Problems”. They are so named because Carroll designed them in bed to distract himself from anxious thoughts while falling asleep. He wrote that as he tossed about in bed, he had two choices: “either submit to the fruitless self-torture of broaching a disturbing topic over and over again, or dictate to himself something absorbing enough to maintain the worry in mind”. bay. A math problem East, for me, such a topic…” I personally identify with Carroll’s situation. Most nights of my life, I fall asleep thinking about a puzzle and have found it to be 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 if you’re still working on this puzzle!
Puzzle #4: Lewis Carroll’s pillow problem
You have an opaque bag containing a marble that has a 50/50 chance of being black or white, but you don’t know what color it is. You take a cue ball out of your pocket and add it to the bag. Then you shake the two marbles in the bag, reach there and take out one at random. He happens to be white. What are the chances that the other marble in the bag is also white?
Don’t let the simple setup fool you. This puzzle is famous for challenging people’s intuitions. If you’re having trouble breaking it, think about it as you fall asleep tonight. That might at least ease your worries.
We will post the solution next Monday with a new puzzle. Do you know of a big puzzle that you think we should cover here? Send it to us: firstname.lastname@example.org
Puzzle Solution #3: Calendar Cubes
The last weeks puzzle asked you to design a pair of working calendar cubes. Remember that a cube only has six faces. Each month has an 11th and 22nd day, so the numbers 1 and 2 must appear on both cubes, otherwise those days could not be rendered. Note that both cubes also need a 0. This is because the numbers 01, 02, … and 09 all need a representation, and if only one cube had a 0, there would be no enough faces on the other cube to accommodate the nine of the other digits. This leaves us with three unoccupied faces on each cube, for a total of six additional points. However, there are still seven numbers that need a house (3, 4, 5, 6, 7, 8 and 9). How can we squeeze seven numbers into six faces? The thing is, a 9 is an inverted 6! Beyond this awareness, several missions work. For example, put 3, 4 and 5 on one cube and 6, 7 and 8 on the other. When the 9 arrives, flip that 6 and, by the skin of our teeth, we have all the dates covered.
There is an economy to this solution that I find beautiful. Two cubes lack space for the task, and yet we creak, exploiting a bizarre symmetry in our numbers. Some might find this fanciful, but that’s 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 bankrupt.
There are two natural extensions of the calendar cube puzzle to other date information. Surprisingly, this theme of hair breadth efficiency persists through them. 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 the same cube face to distinguish them: “Tu” and “Th”. Likewise with Saturday and Sunday, which we will represent by ‘Sa’ and ‘Su’. Monday, Wednesday and Friday have no conflicts so ‘M’, ‘W’ and ‘F’ will do. We find ourselves in a familiar enigma. We have seven symbols to fill on just six sides of a cube. Do you see the solution? The God of Symmetry honors us again, letting ‘M’ represent Monday and, in reverse, Wednesday.
We have months left, which I gave you as an extra challenge last week. Can we expose all three letter month abbreviations: ‘jan’, ‘feb’, ‘mar’, ‘apr’, ‘may’, ‘jun’, ‘jul’, ‘aug’, ‘sep’, ‘ oct’, ‘nov’ and ‘dec’, with three other cubes containing lowercase letters? There are 19 letters involved in an abbreviation of months: ‘j’, ‘a’, ‘n’, ‘f’, ‘e’, ’b’, ‘m’, ‘r’, ‘p’ , ‘y’ , ‘u’, ‘l’, ‘g’, ‘s’, ‘o’, ‘c’, ‘t’, ‘v’, ‘d’, again exactly 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 fit each 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 shown below:
Cubic 1 = [j, e, r, y, g, o]
Cubic 2 = [a, f, s, c, v, (n/u)]
Cubic 3 = [b, m, l, t, (d/p), (n/u)]
Somehow, the few symmetries of our numbering and lettering systems perfectly allow the construction of calendar cubes for days, weeks and months, leaving no room for maneuver.
You may be wondering: if there are 19 letters for 18 boxes, why is it not enough to combine only the ‘u/n’ pair or the ‘d/p’ pair? It seems either would save the extra slot. The rest of the article answers that question and gets a little involved, so stick around only if you’re curious about the answer and don’t want to figure it out on your own. The reason is that if ‘d’ and ‘p’ were split 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 only ‘d’ and ‘p’ share a face while ‘u’ and ‘n’ do not. The abbreviation for June insists that ‘j’, ‘u’ and ‘n’ are on different cubes:
Cubic 1 = [j, …]
Cubic 2 = [u,…]
Cubic 3 = [n,…]
Also, ‘a’ must share a cube with ‘u’ to form ‘jan’:
Cubic 1 = [j, …]
Cubic 2 = [u, a, …]
Cubic 3 = [n,…]
But then how do you do ‘aug’? The letters “a” and “u” share a face. The only way out is to also use ‘u/n’ symmetry.
Let us know how you completed this week’s challenge in the comments.