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5 questions featuring probability & other concepts

1. How many rectangles are there in a line of fifteen squares where sides are lined up with the adjacent sides touching the entire length of the side?

2. If all the diagonals of a regular pentagon are drawn, how many triangles are formed?

3. If a 75-degree sector of a circle rotates around the center of a circle in 75-degree movements, how many sector movements will be needed for the sector to land directly in its original position?

4. A steel band is fitted around the equator. The band is removed and cut, and an additional 10 feet is added. The band now fits more loosely than it did before. How high off the ground is the band?

5. Using straight lines, connect these nine dots with the fewest number of lines without raising your pencil.

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Solution Preview

1. Imagine 15 squares laid out side by side and numbered 1-15 from left to right.
How many 1-square rectangles are there? Obviously 15.
How many 2-square rectangles are there? The first one starts at square 1 and the last one starts at square 14. Therefore there are fourteen 2-square rectanges.
How many 3-square rectangles are there? The first one starts at square 1 and the last one starts at square 13. Therefore there are thirteen 3-square rectanges.
Continue in this vein until you count two 14-square rectangles and one 15-square rectangle.
How many total rectangles are there? The answer is 15 + 14 + 13 + 12... + 3 + 2 + 1 = 120 total ...

Solution Summary

Answers to a 5-question quiz on probability, geometry and calculation. Concepts covered include:
1. Counting large numbers of possibilities
2. Triangles formed by connecting the vertices of a regular polygon
3. Calculating the movement of a sector around a circle
4. The cirfumference of a band around the Earth
5. Connecting an array in the minimum number of moves