Twelve easy pieces:
The puzzle and some activities
Twelve easy pieces is a moderately difficult puzzle that is shown in picture one.
Picture 1
It is a 6 inch by 8 inch rectangle dissected into twelve polygons, each having an area of 4 square inches. The principle of its design is shown in picture two.
Picture 2
Activities for elementary grades.
1. Construction of the puzzle.
Students measure, draw, and cut out the pieces according to instructions. They play with the pieces, making shapes on their own, and shapes suggested by the teacher.
To draw the puzzle you need a ruler and an index card (for drawing right angles).
a. Make an eight inch by 6 inch rectangle and draw its diagonal.
b. The two right triangles that you drew have a common hypotenuse with length 10 inches. Make tick marks along the hypotenuse at 2 inches and 6 inches, dividing it into segments of length 2, 4, and 4 inches.
c. On the top horizontal side of the rectangle, make a tick mark at 4 inches (the midpoint).
d. On the bottom horizontal side, make tick marks at 2 inches and 6 inches, dividing it into segments of length 2, 4, and 2 inches.
e. On the left vertical side, make a tick mark two inches from the bottom. Do the same thing on the right vertical side.
f. Now finish the interior lines as shown below:
Picture 3
2. Can you find the lengths of the remaining sides of the pieces?
You will need to use the Pythagorean Theorem (a2+b2=c2) in many of the calculations. In picture 4 we show the lengths.
Picture 4
3. Which pieces are the same?
Use two concepts. When pieces are made from one-color poster board, "same" means "congruent". But if two-color poster board is used, mirror reflections of some pieces are not congruent to the originals.
4. Classification of the pieces.
4 congruent right triangles (not all the same).
3 congruent right isosceles triangles (all the same)
2 other triangles (not the same).
1 long kite.
1 short kite.
1 square.
5. Measurements.
Measurements of length can be done in inches (and 1/16s of an inch), and/or in centimeters and millimeters. After placing tick marks on the sides of the rectangle and on its diagonal, you can compute all remaining side lengths without measuring them! See Picture 4, where we have left radicals in the measurements. Angles are measured in degrees. Each student should make a data sheet with the results.
6. Areas.
It should be shown that all pieces have the same area, namely 4 square inches. The methods for doing this: "cut and paste", and possibly the area of a triangle. Also, each half of the puzzle has area 24 square inches, so if you are able to get areas of 5 pieces (or four, if the remaining two are congruent) on one side of the diagonal, you may get the area of the other piece(s) by subtraction.
Advanced activities.
1. Construction.
Can you find the radius of a circle inscribed in a 6,8,10 triangle?
This can be used as a step in the construction of the puzzle. (Here is a proof of the fact that the radius of a circle inscribed in any triangle with area K and sides a, b, and c (and thus semi-perimeter s = (a + b + c)/2) is K/s. From this fact it is easy to see that for a 6, 8, 10 triangle, the radius of the inscribed circle is 24/12=2.)
2. Description of the pieces.
2.1 Symmetries (rotations and reflections).
2.2 Measurements obtained both theoretically from a 3, 4, 5 triangle, (calculator: TI-34 II), and experimentally. (Pythagorean theorem, trigonometric functions.)
2.3 Areas and perimeters of all pieces.
2.4 Making similar figures from pieces.
Click on the Flash animation below to view step-by-step how to construct the twelve pieces:
3. Free design.
Each student is asked to present two original designs. (They have to give a drawing with a scale 1:1). A design doesn't have to utilize all the pieces, and there are no restrictions on the arrangement.
Here are 12 designs we made.
1. star and angel
2. ship
3. plesiosaur
4. hawk and dove
5. flower 1
6. flower 2
7. doll
8. candle
9. bird
10. animal 1
11. animal 2
12. abstract design