I decide to write an easy book called "List of All the Brooklyn Public Library Books which do not Contain Their Own Title." The Brooklyn Public Library agrees to help and generously offers to keep my book as soon as it is written.

Very proud, I carve my title beautifully on the cover. Naturally, I then go through each book in the Library and either write their title in my book or do not, depending on whether their title does not or does appear inside that book. Many years later, I am almost finished, and I reject the last one on the shelf, "ZZZZZZ…, the Art of Sleeping Forever," because it contains its own title. Just as I am about to declare the book finished, I realize that I have to decide if I should include my own book in the list.

Right now, the title is not in the book, therefore this is a book that does not contain its title, which means that it should be added to the list. But if I do add it to the list, then the title appears inside the book and my book does not belong in the list! I am stuck! I can neither write the name of my book nor omit it! I ask many logicians but none of them can help: in fact they tell me, this is a famous flaw in set theory discovered by Bertrand Russell at the beginning of the Century. Mathematicians and philosophers are still working on it (see last Tuesday's Science Times). I cry a lot, throw the book in the garbage and decide to write a much easier book: The List of All The Things I Cannot Write...

There are 37 children in the class and I want to organize a tournament to find the strongest Go player: I setup 18 matches and choose one child at random to stand by. After round 1, 18 kids are eliminated and I have 19 left. I must now setup 9 matches and have one kid stand by. After round 2, 9 kids are eliminated and I have 10 left. How many matches have been played when I declare the winner? After you find the answer, do you think there is another (shorter) way to get it? Suppose there were 231 children in the school competing.

Let's try another adding job: Let's add all the numbers from 1 to 10. After we have the answer (55), let's try all the numbers from 1 to 20. Clearly, you can add the numbers and find 210, but can we cheat, can we find a method to make it simple? Also, when I ask for the sum of all the numbers from 1 to 1,000 we better use the trick.

Once you have computed the answer (18+9+5+2+1+1=36), do you think you could answer the same question if there were 61 kids in the class? Assuming someone guesses right (the answer is 60), we need to justify it. Why, if there are 61 kids, does it take 60 matches to find the winner? What we are looking for is the fact that each match eliminates just one kid.

The answer: let us write the same sum twice (the second time backwards):

1 + 2 + 3 + . . . + 998 + 999 + 1000

1000 + 999 + 998 + . . . + 3 + 2 + 1

Now you might notice that each column adds up to 1001. Since there are exactly 1000 columns, the sum of all the numbers above is 1,001 times 1,000 =1,001,000. The number we wanted is just half of that.

Finally a puzzle to keep you busy till next week (from a wonderful book called Problem Solving Through Recreational Mathematics).

Please note that each of the 3 cars can attach at either end.

Hint: You want to go from ALB to BLA,

but start by trying to go from ALB to LAB.

A set is a collection of objects. For instance, the set of children in the class, the set of books on the table, the set of pink elephants in the room. Wait, there is none! That's OK, we call that set the empty set, the set that contains nothing (we write it up as the greek letter phi). It is identical to the set of live butterflies who are 200 years old. In general, we say that the objects belong to the set or are elements of the set. We also say that the set contains elements or objects. There is no rule that restricts elements from being anything, including sets (a recursive concept).

Then we define the union of two sets (the set of elements which belong to one of the two sets), the intersection of two sets (the set of elements which belong to both sets), etc…

We also separate the sets into finite sets and infinite sets depending on the number of their elements: the set of children in the school is finite, the set of all the numbers 1, 2, 3, etc … is infinite.

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