Alice in Puzzleland

D. Indumathi, The Institute of Mathematical Sciences, Chennai

Have you read Alice's Adventures in Wonderland? Written more than 150 years ago by Lewis Carroll, it is the highly imaginary and imaginative story of a young girl called Alice who dreams that she has followed a talking rabbut through a rabbit hole and finds a fantastic world peopled by strange and peculiar creatures. If you haven't read it, you must try and find a copy of the book.
The story is appealing to both children and adults; one unusual theme is that Carroll uses logic to make sense of the fictional world. Carroll later wrote a sequel called Through the Looking Glass where Alice climbs through a mirror into a world of opposites, still with its own strange logic.
The sketch of a Google "doodle" shows Alice as the "L" in Google, and was published in February this year as a tribute on the 200th birthday of Sir John Tenniel. Tenniel had beautifully illustrated the book of Alice. The famous Cheshire cat that can disappear, leaving only its smile behind, is also shown. Also see one of Tenniel’s illustrations of a caterpillar using a hookah.
Several writers have been inspired by "Alice". One of the most famous is the logician and popular writer Raymond Smullyan. Smullyan is well known for his popular books on logic puzzles and he has based his entire book Alice in Puzzleland" on the characters in "Alice", especially the White Queen and the Red King (who are all from a pack of cards!)
Read on for a few of the puzzles from this book. The first three are simpler algebra and the logic puzzles follow.
Let's start with an easy one. How much is a million divided by a quarter? (Ans 1 below).
A bottle of soft drink costs Rs 30. The soft drink costs Rs 26 more than the bottle. How much is the bottle worth? (Ans 2 below).
A certain farmer had no money to pay his taxes. So the tax collector took one tenth of his land away from him. After the land had been take, the farmer had 10 acres left. How much land did he originally have? (Ans 3 below).
Tweedledum and Tweedledee had a bet. Whoever won the bet would get one baby rattle toy to add to their collection of rattles. Tweedledum realised that if he loses the bet, he will have the same number of rattles as Tweedledee. If he wins the bet, then he willhave twice as many rattles as Tweedledee. How many rattles does each one have? (Ans 4 below).
The White Queen said, "Whenever the Red King is asleep, everything he believes is wrong. On the other hand, everything he believes while he is awake is true. Well, last night at ten o' clock sharp, the Red King believed that both he and the Red Queen were asleep at that time. Was the Red Queen asleep or awake at the time?" (Think carefully about what is being asked. See Ans 5 below).
The Red Queen said, "I am like the Red King. I also believe only false things when I am asleep and believe only true things when I am awake. Now, last night at 11 o' clock, the Red King believed I was asleep. At the same time, I either believed that he was asleep or I believed that he was awake. Which did I believe?" (Ans 6 below).
The White Queen said, "Here is a true story. I once had to post four letters. I had written then, and had the four envelopes correctly addressed, but I was careless and put some of the letters into the wrong envelopes. However, I put only one letter in each envelope. As it happens, I either got three of them exactly right, or I got exactly two of them right, or I got exactly one of them wrong. How many did I get right?" (Ans 7 below).

Answers to the puzzles
Ans 1. A million divided by a quarter is 4 million (not a 1/4 million since you are dividing, not multiplying).
Ans 2. Let the bottle cost B=Rs X. Then the drink D costs Rs 26 more, or D=26+X. The bottle and drink together cost Rs 30. So B+D=30, or X+(26+X)=30, or 2X=4, or X=2. The bottle costs Rs 2.
Ans 3. Let the farmer have X amount of land. One tenth of this is taken away, so (9/10) of it is left. This equals 10 acres, or (9/10)X=10, or X=100/9 or the farmer had 11 and 1/9 acres to start with.
Ans 4. Let Tweedledum have X rattles and Tweedledee Y rattles. If Tweedledum loses, Tweedledee gets an extra rattle, so he will have (Y+1) rattles, and this will equal what Tweedledum has. So X=Y+1. If Tweedledum wins, he gets the extra rattle and this will be twice what Tweedledee has, or X+1=2Y. Replace X=Y+1 from the earlier statement to get (Y+1)+1=2Y, or simplifying, Y=2. So X=Y+1=3.
Ans 5. If the King was awake, whatever he believes is true. So he could not have believed that both he and the Queen were asleep. So he must have been asleep. But then whatever he believed must be false. He believed that both of them were asleep and he was in fact asleep. So the only way for this statement to be false is if the Queen was awake. So the Queen was awake.
Ans 6. This is a little more complicated. We have to analyse the cases for both the King being asleep or awake and the Queen being asleep or awake. Let's start with the King. He believes that the Queen was asleep. Suppose he was asleep but since he is asleep this belief is false. So the Queen was actually awake. On the other hand, if he was awake, then what he believes is true. So the Queen was actually asleep.
With this information in mind, let us analyse the Queen's beliefs. She either believes that (1) the King is awake or she believes that (2) the King is asleep. She does this while either asleep or awake.
Let us start with the possibility that the Queen is asleep. Suppose she believes (1), that the King is awake. Since she is asleep, this is false, so the King is actually asleep. So what the King believes (that the Queen is asleep) is false. So the Queen is awake, but that contradicts our starting point that the Queen is asleep. So this combination is ruled out. What if she believes (2), that the King is actually asleep. She herself is asleep, so this belief is untrue. So the King is actually awake, and when he is awake, his belief (that the Queen is asleep) is true and this is consistent with our starting point that the Queen is asleep. 
Now let us consider the possibility that the Queen is awake. Suppose she believes (1), that the King is awake. Since she is awake, this belief is true, so the King is indeed awake. Hence the King's belief (that the Queen is asleep) is true but this contradicts our starting point that the Queen is awake. So let us try the other belief, that she believes (2), that the King is asleep. Since she is awake, this belief is true. So the King's belief (that the Queen is asleep) is false, and so the Queen is awake, which is consistent with our starting point.
So in the two cases we have analysed that are consistent,
. the King is awake and the Queen is asleep, or
. the King is asleep and the Queen is awake.
In the first case, the Queen is asleep and her belief is untrue. Since the King is awake, she must believe the opposite, that the King is asleep. 
In the second case, the Queen is awake and her belief is true. Since the King is asleep, she must believe that the King is asleep.
Have you noticed: in both cases, the Queen believes that the King is asleep!
Ans 7. This is simple. If you had put three letters correctly in their envelopes, then the last letter has to go into the last envelope so it is not possible to get three right and one wrong. Similarly, getting exactly one wrong means getting three right, and we just saw that this is impossible. So she must have got exactly two of them right and the other two were swapped.
From Alice in Puzzleland, Raymond Smullyan.