Math Delights

By Henri Picciotto, adapted slightly from www.picciotto.org/math-ed/angles/angles-around.html

Place just enough different pattern blocks around a point, so that a vertex (corner) of each block touches the point, and no space is left between the blocks.  For example, with 3 blue rhombuses or 3 yellow hexagons, you can make these figures:

(These images are from www.europa.com/~paulg/mathmodels/angle_meas.html.)

1. Use the following chart to keep track of your findings.
   • Every time you find a new combination, circle the appropriate number on the list below and draw it on your answer sheet.
   • Cross out any number you know is impossible.
   Since the all blue three-blocks solution and the all yellow three-blocks solutions are shown above, circle the 3 in the all blue row and the 3 in the all yellow row.
   
   Figure out how many all tan small rhombuses you need to surround a vertex (corner) with other small rhombuses of the same shape. The middle figure above shows that you'll need more than 3.
   

   Colors	How many blocks you used
   All Blue	3	4	5	6
   All Green	3	4	5	6
   All Orange	3	4	5	6
   All Red	3	4	5	6
   All Tan	3	4	5	6	7	8	9	10	11	12
   All Yellow	3	4	5	6
   Two Colors	3	4	5	6	7	8	9	10	11	12
   Three Colors	3	4	5	6	7	8	9	10	11	12
   Four Colors	3	4	5	6	7	8	9	10	11	12
   Five Colors	3	4	5	6	7	8	9	10	11	12
   Six Colors	3	4	5	6	7	8	9	10	11	12

2. How many solutions are there altogether?
3. Which blocks offer only a unique solution? Why?
4. Why are the tan block solutions only multiples of 4?
5. Explain why the blue and red blocks are interchangeable for the purposes of this activity.
6. Describe any systematic ways you came up with to fill in the bottom half of the chart.
7. How do you know that you have found every possible solution?
8. Which two- and three-color puzzles are impossible, and why?
9. Which four-color puzzles are impossible, and why?
10. Why is the five-color, eight-block puzzle impossible?
11. Which six-color puzzles are impossible, and why?