Problem Solving and Computing (CSC-103 98S)

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# Problems 1: Introductory Problems

Assigned: Tuesday, January 20, 1998
Due: Tuesday, January 27, 1998

Start working on these problems in class, then finish your "resolutions" before next Tuesday. You may work with others on this assignment (and most assignments), but each person should write up his or her own resolutions.

Turn in resolutions for 1, 2, and 7.

1. Paper Strip

[From Thinking Mathematically (page 4).]

Imagine a long thin strip of paper stretched out in front of you, left to right. Imagine taking the ends in your hands and placing the right hand end on top of the left. Now press the strip flat so that it is folded in half and has a crease. Repeat the whole operation on the new strip two more times. How many creases are there? How many creases will there be if the operation is repeated 10 times in total?

2. Patchwork

[From Thinking Mathematically, p. 11.]

Take square and draw a straight line right across it. Draw several more lines in any arrangement so that the lines all cross the square, and the square is divided into several regions. The task is to color the regions in such a way that adjacent regions are never colored the same. (Regions having only one point in common are not considered adjacent.) What is the fewest number of different colors you need to color any such arrangement?

3. The Beans Game

[From the notes of Emily Moore with modifications by Samuel A. Rebelsky.]

We start with 17 beans in a pile. Two players alternate, each removing 1, 2, 3, or 4 beans in a turn. The player who removes the last bean wins. What strategy should the first player use to guarantee that (s)he will win? Will this strategy work for other size piles? Are there any size piles for which the second player has a winning strategy?

4. Subtract-a-Square

[From the notes of Emily Moore.]

In the game ``Subtract-a-Square,'' a positive integer is written down and two players alternately subtract squares from it until a player is able to leave zero, in which case he/she is the winner. What square should the first player subtract if the original number is 29?

5. Language Students

[From the notes of Emily Moore.]

A high school offers classes in Spanish and French. There are 200 students; 81 take Spanish and 93 take French. If 20 percent of the students take no language, how many take both Spanish and French?

6. Sandwiches

[From the notes of Emily Moore.]

A deli has made 1000 sandwiches for a party. Two-thirds of the ham sandwiches also have cheese, and three-quarters of the cheese sandwiches also have ham. If there are 120 sandwiches that have neither ham nor cheese, how many sandwiches have both ham and cheese?

7. Four Fours

[From the notes of Emily Moore, with modifications by Samuel A. Rebelsky.]

Using mathematical symbols to combine four fours it is possible to write expressions for all the numbers from 0 to 100, as well as millions of others. For example,

• `2 = 4/4 + 4/4`.
• `3 = (4+4+4)/4`.
• `64 = (4+4) * (4+4)`.

Using any symbols that do not themselves contain digits, see how many numbers you can form using four fours. Some should be easy. Some people find these numbers particularly difficult: 13, 19, 33, 85.

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