Join the Dots


 

   Question:

Can you connect 9 dots laid out 3x3 using 4 straight lines, without lifting your pencil from the paper?

   Suggested Materials:

paper
pencil
4 sticks of uncooked spaghetti and 9 pennies

or 4 toothpicks and 9 Cheerios or other flat breakfast cereal
or whatever other manipulative is handy to simulate the lines and dots

   Procedure:

  1. Place 9 "dots" on your desk or table.
  2. Using the 4 sticks experiment with connecting the dots using 4 straight lines. (Or just use pencil and paper.)

   Questions to ask yourself as you work:

  1. Can you do the puzzle if the lines do not cross each other?
  2. Must your lines stay within the grid of dots?
  3. What if you extend a line past the edges of the arrangement of dots?

A hint...

Join the Dots: Hint


 

Here is a hint for you before you look at the solution:

Try extending one of the lines past the edges of the dots.

 

   Solution

Join the Dots: Solution


 

  Did you get this solution?

If you found any others we'd love to hear about them.

Rosebush Puzzle 1*

           

   Question:

A gardener laying out a bed of roses finds that she can plant 7 rosebushes so that they form 6 straight lines with 3 rosebushes in each line.

How is this possible?

   Suggested Materials:

paper
pencil
6 sticks of uncooked spaghetti and 7 pennies

or 6 toothpicks and 7 Cheerios or other flat breakfast cereal or whatever other manipulative is handy to simulate the lines and rosebushes

   Procedure:

  1. Place 7 "rosebushes" on your desk or table.
  2. Using the 7 sticks experiment with arranging the "bushes" into 6 lines of 3 each.

   Questions to ask yourself as you work:

  1. Will there be 3 rose bushes in each row if you have only 7 bushes and you place the six rows parallel to each other?
  2. If your rows are not parallel, how else could they be arranged?
  3. If your rows intersect, where should you place one of your rose bushes?

    Check for understanding

Rosebush Puzzle 1


When you have placed the first two sticks and one bush
the puzzle will look something like this:

 

  1. After placing the first rosebush, how many do you have left?
  2. How many more rosebushes need to be placed in order to have 3 in that row?

A hint...

Rosebush Puzzle 1: Hint


Here are a couple of hints for you before you look at the solution:

Try arranging the rosebushes in a circular pattern.

Place two sticks crossed so that three rosebushes fall on them.

   Solution

Rosebush Puzzle 1*

           

Did you get this solution?

If you found any others we'd love to hear about them.*

*Here's another solution sent to us by Hans Fiedler:


    Rosebush Puzzle 2* 

   Question:

A gardener laying out another bed of roses planted 10 rosebushes in 5 straight lines with 4 bushes in each line.

How did she do it?

   Suggested Materials:

paper
pencil
5 sticks of uncooked spaghetti and 10 pennies


or 5 toothpicks and 10 Cheerios or other flat breakfast cereal
or whatever other manipulative is handy to simulate the lines and rosebushes

   Procedure:

  1. Place 10 "rosebushes" on your desk or table.
  2. Using the 5 sticks experiment with arranging the "bushes" into 5 lines of 4 each.

   Questions to ask yourself as you work:

  1. If there are 4 bushes on each of the 5 lines and the lines are parallel, how many bushes would there be in all?
  2. Since that would be 20 bushes and you only have 10 bushes to place, can the lines be parallel?

    Check for understanding

     Rosebush Puzzle 2 Check  

Imagine the shape of a star...

 

 

  1. When you place 4 rosebushes on one of the lines, how many bushes do you have left?
  2. How many rosebushes are toward the inside, and how many are toward the outside?

A hint...

Rosebush Puzzle 2 Hint

Here's a hint for you before you look at the solution.

Try starting with an irregularly spaced inner circle of 5.

OR

Draw the four rosebushes across the top horizontal line of a star.

 

   Solution

  Rosebush Puzzle 2*: Solution

Did you get this solution?

Here's an alternate solution by Latoya Benjamin of Houston, Texas.

Thanks, Latoya!

  Rosebush Puzzle 2*: Solution

More solutions from Sean Peters

Here are a two more solutions to the ten points in 5 rows of 4 problem.



And here's an observation: no point can lie on three (or more) of the rows. Consider that if it does, then you have used all ten points to construct those lines, call them A, B, and C:

Assume there is another row of 4 points, call it D. By the pigeonhole principle, it must contain at least 2 points from one of the 3 lines A, B, or C. Without loss of generality, assume that D contains at least 2 points that lie on A; then A and D are collinear, a contradiction.

Note that the illustration above isn't the only way that 3 lines can all share a point, but the logic still holds for all cases.

I have strong heuristic evidence that these 6 are the only solutions but I haven't yet written a rigorous proof. If I do complete a proof, you will be among the first to know.

  Rosebush Puzzle 2*: Solution

Six solutions from Sean Peters

Here are all six of the solutions to the ten points in 5 rows of 4 problem.

http://mathforum.org/k12/k12puzzles/ diakses 01 mei 2007

 

Rosebush Puzzle 3*

   Question:

A gardener laying out a third bed of roses planted 19 rosebushes in 9 straight lines with 5 bushes in each line

How did she do it?

   Suggested Materials:

paper
pencil
9 sticks of uncooked spaghetti and 19 pennies

or 9 toothpicks and 19 Cheerios or other flat breakfast cereal
or whatever other manipulative is handy to simulate the lines and rosebushes

   Procedure:

  1. Place 19 "rosebushes" on your desk or table.
  2. Using the 9 sticks experiment with arranging the "bushes" into 9 lines of 5 each.

   Questions to ask yourself as you work:

  1. If there are 5 bushes on each of the 9 lines and the lines are parallel, how many bushes would there be in all?
  2. Since that would be 45 bushes and you only have 19 bushes to place, can the lines be parallel?

    Check for understanding

Rosebush Puzzle 3: Check

Imagine the shape of a six-pointed star...

 

  1. When you place 5 rosebushes on one of the lines, how many bushes do you have left?
  2. How many rosebushes are toward the inside, and how many are toward the outside?

A hint...

Rosebush Puzzle 3: Hint

Here's a hint for you before you look at the solution.

Try starting with an irregularly spaced inner circle of 12.

OR

Draw the five rosebushes across the top horizontal line of a six-pointed star.

 

   Solution

Rosebush Puzzle 3: Solution

Did you get this solution?

Visitor Jason Christianson describes this alternate solution:

Mark the three vertices of an isosceles triangle. Draw the two legs from the top vertex to the bottom two vertices. Pick three points on one leg; reflect those points across the altitude and onto the opposite leg. Now construct three segments connecting each bottom vertex to those three new points on the opposite leg. Last, drop a segment straight down from the top vertex, passing through all the points of intersection. Mark the point at the end of this line, and at all the intersections.

Download a GSP sketch (4K) to experiment with this solution.


 

M Puzzle*

   Question:

Can you construct nine triangles by drawing three straight lines through a capital M?

   Suggested Materials:

paper
pencil
3 sticks of uncooked spaghetti


or 3 toothpicks
or whatever other manipulative is handy to simulate the lines

   Procedure:

  1. Draw a large M on your paper.
  2. Use the 3 sticks and try various ways to place them across the M.

   Questions to ask yourself as you work:

  1. Are any triangles formed if you place one of the sticks vertically?
  2. Are any triangles formed if you place one of the sticks horizontally?
  3. What is the maximum number of triangles you can make by placing just one stick?

    Check for understanding


 

M Puzzle Check

When you have placed the first stick
the puzzle will look something like this:

 

 

  1. When you have placed one stick, look at the triangles that you have made. How many more triangles do you still need to make?
  2. You have 2 sticks left. Can you place them so that you will cross existing triangles and thereby double the number of triangles?

A hint...

 

M Puzzle Hint

Here's a hint for you before you look at the solution.

When you have placed the last two sticks you may see a shape within the M.


 

   Solution

     

M Puzzle Solution

Did you get this solution?

 

But wait! What if we look for overlapping triangles? FlwrGrl89 writes that we've made more than nine triangles with our three sticks. For example...

     

Kris' M Puzzle Solution

 

 

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