Problems 0-9

Problem 0

In the following configuration of numbers (commonly known as Pascal's triangle) each row as a 1 at either end, and the interior numbers are each the sum of the two numbers above:

          1
         1 1
       1  2  1
      1 3   3 1
     1 4  6  4 1
    1 5 10 10 5 1
          .
          .
          .

Counting the first row (containing a single 1) as Row 0, how many numbers in Row 1,000,000 are odd? (The obvious generalization is to determine the number of odd numbers in Row N.)

Problem 1

Assume you buy a chocolate bar that's marked into 28 smaller pieces in four rows of seven pieces:

    |----------------------------------|
    |    |    |    |    |    |    |    |
    |----------------------------------|
    |    |    |    |    |    |    |    |
    |----------------------------------|
    |    |    |    |    |    |    |    |
    |----------------------------------|
    |    |    |    |    |    |    |    |
    |----------------------------------|

Your task is to break the bar into the 28 pieces with the fewest number of straight line breaks without stacking. Clearly you can do it with 27 breaks (three to separate the three rows, and then six per row to separate the pieces in that row). Can you do it in fewer breaks, or can you come up with an argument for why it can't be done in fewer than 27 breaks?

Problem 2

This is somewhat similar to Problem 1. This time you're given a cube of wood and asked to cut it into 27 equal sized smaller cubes (see diagram). Here, though, you're allowed to stack, so after making the first cut, you can stack the resulting two pieces before making the second cut. Clearly you can do the job without stacking in six cuts; can you take advantage of stacking to do it in fewer than six cuts? If you don't think you can, can you come up with a convincing argument for why you can't?

Problem 3

This is the same as Problem 2, except the original wooden cube is hollow, i.e., the middle small cube is missing. You now need to cut the cube into 26 equal sized smaller cubes, and the question is again to determine the fewest cuts.

Problem 4

Pat and Chris recently attended a party with three other couples. At the party various handshakes were exchanged. No one shook hands with their own companion and, not surprisingly, no one shook hands with themselves. After all the handshaking was finished, Chris asked each person, including Pat, how many people they shook hands with. Surprisingly, each person gave a different answer. How many hands did Pat shake?

Problem 5

Two poles with height a and b are distance d apart (along level ground). From the top of each pole there is a guy wire stretching down to some common point P. Where should P be to minimize the amount of wire needed?

Problem 6

n points are chosen on a circle, and each pair of points is joined by a line segment. Assuming no three of these segments intersect at a common point inside the circle, how many points of intersection are there?

Problem 7

Given a positive integer n, how many quadruples of numbers (a, b, c, d) are there such that

Problem 8

Derive the operations + (plus), * (times), and / (divide) from - (minus) and reciprocal.

Problem 9

Invent a single binary operation which you can use to derive + (plus), - (minus), * (times), and / (divide).

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