The illustration below shows a way to place three squares of paper flat on a table so that each sheet touches every other sheet. A fourth sheet can be positioned on top, and every sheet will still touch every other sheet.
1. Cut out six square sheets of paper, all the same size. Can you arrange them in two layers so that every sheet touches every other sheet? A corner can touch a side, but two corners touching don’t count. You may not bend or fold the sheets of paper.
2. Suppose you are allowed to have three layers of square sheets. What is the maximum number of sheets that can be arranged so that every sheet touches every other sheet? You may press down on the papers so that sheets in nonadjacent layers can touch.
3. If you used as many layers of paper as you wanted, what is the maximum number of sheets of paper that can be arranged so that every sheet touches every other sheet? Again, you are allowed to press down.
Queens At Peace
1. On this chessboard, four white queens and four black queens are placed so that no queens of opposite color are threatening each other. Can you place eight white and eight black queens so that no queens of opposite color threaten each other? (Two queens threaten each other if they are in the same row, column, or diagonal.)
2. Can you do the same with eight white and 10 black queens? The solution should be symmetrical.
3. Now try to do the same thing with nine white and nine black queens. This solution needn’t be symmetrical.
4. How about nine white and 10 black queens? The solution does not have to be symmetrical. Is it possible with eight white and 11 black queens, or with 10 white and 10 black queens?
Polygons, like living things, can be organized by family, genus, and species. Some members of the quadrilateral class are subclasses of other types of quadrilaterals. Squares, for example, are a subclass of rectangles, because a square is a rectangle whose sides are all the same length. Rectangles, however, are not a subclass of squares, because not all rectangles are squares.
One quadrilateral may be a subclass of more than one other kind of quadrilateral. Squares are a subclass of both rhombuses and rectangles. But since most rhombuses are not rectangles, and most rectangles are not rhombuses, neither is a subclass of the other.
Below, read about a dozen types of quadrilaterals—by definition, all have four sides, but each has unique qualities. Then answer the questions on the right.
| A. Convex quadrilateral: a quadrilateral whose angles are all less than 180 degrees. |
B. Cyclic quadrilateral: a quadrilateral whose corners all lie on a single circle.
C. Kite: a quadrilateral with two pairs of adjoining sides of equal length.
D. Parallelogram: a quadrilateral whose opposite sides are parallel.
E. Quadrilateral: a flat polygon with four sides.
F. Rectangle: a quadrilateral with four 90-degree angles.
G. Rhombus: a quadrilateral whose sides are all the same length.
H. Square: a quadrilateral with four sides of equal length and four 90-degree angles.
I. Trapezoid: a quadrilateral having only two parallel sides.
J. Unnamed quadrilateral 1: has opposite angles of equal measure.
K. Unnamed quadrilateral 2: has one pair of opposite angles both 90 degrees.
L. Unnamed quadrilateral 3: has one pair of opposite sides of equal length.
| 1. Which two definitions describe exactly the same type of quadrilateral?|
2. Besides E (Quadrilateral), which two types of quadrilaterals are not necessarily convex?
3. Quadrilaterals of class K (one pair of opposite angles both 90 degrees) are a subclass of which two other types of quadrilaterals, not counting E?
4. Can you find a chain of six of the quadrilaterals in which each type is a subclass of the next? Hint: The first is a square and the last is a quadrilateral.
5. Can you find one set of four quadrilaterals in which no one quadrilateral is a subclass of another? Hint: Try drawing pictures.
Can you define a fifth type of quadrilateral that is neither a subclass nor a superclass of these four quadrilaterals?
Illustrations By Eric Claro Solution
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