## Monday, August 10, 2009

### Dodgson (Carroll)

In my last post, A Common Thread, I posed four puzzles and asked you readers to figure out what they all had in common. The common thread that binds all four puzzles is that they were all originally written by none other than Charles Lutwidge Dodgson, better known by his pen name, Lewis Carroll. (Incidentally, some of you may notice that the title of this blog and the "Now is the time..." quote that I use also come from the works of Lewis Carroll. It would be an understatement to say that I'm a huge fan of his work.)

Charles Dodgson was an English mathematician, logician, and writer. Dodgson was born the son of a vicar at the Old Parsonage, Newton-by-Daresbury, Cheshire. He was educated at the Rugby School, where the student mathematical society is still called the Dodgson Society in his honor, and later at Christ Church Oxford, where he would spend the rest of his life employed as a lecturer.

Dodgson is best known by his pen name, Lewis Carroll. It was under this name that he published his most famous book, Alice's Adventures in Wonderland, in 1865. The story of Alice's adventures was first told by Dodgson to the three daughters of H. G. Liddell, then the dean at Christ Church. Alice's adventures continued in 1871 in its sequel Through the Looking-Glass. His other works include the peoms Jabberwocky and The Hunting of the Snark.

Dodgson's work, including books written for children, is filled with logical puzzles and paradoxes. Many of the logic puzzles that appeared in his work and in personal correspondence are well-known today.

Do not read any further unless you want the solutions to the four puzzles presented in A Common Thread.

Milky Water
Your are given two glasses, one containing exactly 50 tablespoons of milk, the other containing exactly 50 tablespoons of water. You take one tablespoon of out of the milk glass and mix it with the water. You then take one tablespoon of the water/milk mixture and mix it into the pure milk to obtain a milk/water mixture. Are you left with more water in the milk/water mixture or more milk in the water/milk mixture?
Solution (pun intended): There is exactly the same amount of water in the milk/water mixture as there is milk in the water/milk mixture.

Starting with 50 tablespoons of each, you remove one tablespoon from the milk and add it to the water. this leaves 49 tablespoons of pure milk, and 51 tablespoons of water/milk mixture, which is 1/51 milk and 50/51 water. You then take one tablespoon of the water/milk mixture and add it to the milk. This leaves 50 tablespoons of the water/milk mixture, which has a 1/51 ratio of milk to water.

When you add the tablespoon of 50/51 water/milk mixture to the 49 tablespoons of pure milk, you create a mixture that has a ratio of 1/51 water to milk. To see why this is so, consider that the one tablespoon of water/milk mixture has 51 parts, 50 of them water, and one milk. If you divide the 49 tablespoons of pure milk into parts of the same size as the 51 parts in the single water/milk tablespoon, you get

49 x 51 = 2,499

parts of pure milk. When you add the tablespoon of water/milk, you add one more milk part, and 50 water parts, for a mixture that is 50 parts water and 2,500 parts milk, or 1/51 water, 50/51 milk.

Update: Reader roryparle pointed out to me in the comments that my solution is too specific. My solution assumes that the mixtures of milk and water are mixed homogeneously, which is not necessary.
Starting from a mixture of 50 parts water + 1 part milk, and another container of 49 parts milk, you take one part from the mixture. This will have some fraction, f, of a part of milk, and some fraction (1 - f) of a single part of water (to make the whole add up to 1 part of something). The remaining mixture will contain (1 - f) part milk (it had 1 to begin with and you removed f) and (49 + f) parts water (50 - (1 - f) = 50 - 1 + f = 49 + f).

Add the spoon you've just removed to the milk and you'll have 49 + f parts milk (49 that was there + f from the spoon) and (1 - f) parts water (all from the spoon).

This works for the fraction you assumed, f = 1/51, but also for the cases where you get no milk in the second spoon (f = 0), or you miraculously manage to completely separate the milk and water again (f = 1), or anything in between.
Thanks for the correction! It's great to know that people actually read these things. :)

Four to Five

Make a word ladder starting with FOUR and ending with FIVE. (Every step in a word ladder differs by only one letter from the previous step, but each step must be an English word. You may add a letter, remove a letter, or change a letter, but the words from one step to the next must differ by only one letter.)
Solution: This word ladder looks deceptively simple because it contains only four letters, but it is made more difficult by the fact that you need to change one of the letters, the third, at three different points in the ladder (in this particular solution; there are other solutions).

FOUR → FOUL → FOOL → FOOT → FORT → FORE → FIRE → FIVE

Update: Sharp-eyed reader Xetius pointed out in the comments that there is a shorter solution given the rules above.

FOUR → FOR → FIR → FIRE → FIVE

It seems that Dodgson's original rules for word ladders (he invented the game itself, not just this instance) didn't include the rules for adding and removing letters.

Clock Face

A clock face has the same symbol for all twelve hours, and both hands are exactly the same length (there is no second hand, only an hour hand and a minute hand). The clock stands opposite to a mirror. At what time between 6 and 7 o'clock (not inclusive) will the time read exactly the same on the clock as in the mirror?
Solution: This is a problem in visualization and symmetry. The clock face in the puzzle will read the same as its mirror-image when the left and right halves of the clock face are the same. That will be the case when the hands of the clock are in the position shown in the following illustration.

But at exactly what time will that occur? The clock face can be divided into 360 degrees, with the minute hand advancing at a rate of

360 degrees / hour
6 degrees / minute
1/10 degree / second

The hour hand advances at a rate of

30 degrees / hour
1/2 degree / minute
1/120 degree / second

If you draw a vertical line down the center of the clock face, then look at the angle that each hand of the clock advances to get to the symmetric position, you can see that the two angles add up to 180 degrees.

Now if we take X to be the amount of time elapsed, we can build the following formula:

(X * 1/10 deg/sec) + (X * 1/120 deg/sec) = 180 deg

Solving for X, we get:

X * (1/10 deg/sec + 1/120 deg/sec) = 180 deg
X = 180 deg / (1/10 deg/sec + 1/120 deg/sec)
X = 180 deg / (12/120 deg/sec + 1/120 deg/sec)
X = 180 deg / (13/120 deg/sec)

The next step is a little tricky if you haven't taken an algebra class in awhile. In order to get rid of the degrees and be left with an elapsed time, you need to multiply both the numerator and denominator by the reciprocal of the denominator. This is the same as multiplying the entire right-hand side by one.

X = (180 deg * 120/13 sec/deg) / (13/120 deg/sec * 120/13 sec/deg)

Since 13/120 deg/sec and 120/13 sec/deg are reciprocals of one another, multiplying them together results in 1, cancelling out the denominator. We are left with

X = (180 deg * 120/13 sec/deg)

Since we have degrees multiplied by seconds/degree, the degrees cancel and we're left with seconds.

X = 180 * 120/13 sec
X = 21600/13 seconds
X = 360/13 minutes

So the exact answer is 360/13 minutes after 6 o'clock, which is approximately 27 minutes and 42 seconds past six.

Ravens and Writing Desks

Why is a raven like a writing desk?
Solution: Dodgson didn't have a solution in mind at the time that he wrote this riddle, which first appeared in Alice's Adventures in Wonderland (in chapter 7, A Mad Tea Party), but he later came up with:
Because it can produce a few notes, though they are very flat; and it is nevar put wrong end in front!
which appeared in a later printing of the book. Note the alternate spelling of "nevar", which is "raven" spelled backwards.

Another famous puzzler, Sam Lloyd, proposed a clever alternate solution:
Because Poe wrote on both.

Notes:

The photograph of Dodgson above was taken by Oscar Gustav Rejlander, a pioneer in early photography and a friend of Dodgon's. It was Rejlander who inspired Dodgson to take up amateur photography himself.

For more information on the life and work of Charles Dodgson, read Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life, by Robin Wilson.

Kyle Cronin said...

I've never read Alice's Adventures in Wonderland, but this post has inspired me to see if I can get a copy in the near future. I've known for a while that Lewis Carroll was a mathematician, but I didn't know that mathematics was featured that prominently in the story.

As a side note, it was also nice to finally learn the origin of your username (not that I'd ever been curious enough to search):

http://en.wikipedia.org/wiki/Bill_the_Lizard

Xetius said...

For the word ladder, using you rules of "You may add a letter, remove a letter, or change a letter, but the words from one step to the next must differ by only one letter" why can you not use: FOUR - FOR - FIR - FIRE - FIVE. I must admit that I always thought word ladders had to have the same number of letters, but being able to add a letter, remove a letter makes this much simpler.

Xetius

Anonymous said...

The solution to the first puzzle is more general than you imply. Your answer assumes that the milk and water are mixed evenly, but that's not necessary.

Starting from a mixture of 50 parts water + 1 part milk, and another container of 49 parts milk, you take one part from the mixture. This will have some fraction, f, of a part of milk, and some fraction (1 - f) of a single part of water (to make the whole add up to 1 part of something). The remaining mixture will contain (1 - f) part milk (it had 1 to begin with and you removed f) and (49 + f) parts water (50 - (1 - f) = 50 - 1 + f = 49 + f).

Add the spoon you've just removed to the milk and you'll have 49 + f parts milk (49 that was there + f from the spoon) and (1 - f) parts water (all from the spoon).

This works for the fraction you assumed, f = 1/51, but also for the cases where you get no milk in the second spoon (f = 0), or you miraculously manage to completely separate the milk and water again (f = 1), or anything in between.

Bill the Lizard said...

Kyle,
You can usually get Alice's Adventures in Wonderland and Through the Looking-Glass together for the price of one new paperback (around $6 or$7 USD) at most book stores. It would definitely appear on my list of books everyone should read at least once.

Also, I dream of the day when my blog is so popular that it's added as a footnote to that Wikipedia article. :)

Bill the Lizard said...

Xetius,
That's a good solution. Interesting, since Dodgson was the inventor of word ladders, and the rules that I posted, and he also came up with the solution I gave. I wonder if the rules evolved some time after that solution was known? I'll have to investigate. Sharp eye!

Bill the Lizard said...

roryparle,
You're right, the general solution is better. That's two corrections I'll have to make today (see the above comments). Thanks for pointing this out.

Matt said...

I'd argue that the solution to the first puzzle is even yet more general.

After the mixing, you have two jugs, each containing the same volume of liquids. In the edge case you end up with a completely unmixed system, there's no water in the milk, and no milk in the water.

You know that there's milk in the water jug. But since the volume is exactly the same as what it started with, there must have been an equal amount of water displaced. The only place it could have been displaced to is the other jug.

Thus, there is exactly as much milk in the water jug as there is water in the milk jug.

For those interested in (re)reading Alice's Adventures in Wonderland, I'd recommend trying to track down Martin Gardener's "The Annotated Alice", which comprises both Alice books as well as quite a bit of commentary.

Bill the Lizard said...

Matt,
That's a nice, intuitive explanation of the Milky Water puzzle. Thanks for the tip on The Annotated Alice, also. Gardner is one of my absolute favorite writers, but I wasn't aware of this book before.