We’re going to see how many pencils we can put together so that each pencil touches all the others. Try coming up with your own solutions, and then follow these instructions to create some famous patterns.

You will need

two pencils pointing right, under two pencils pointing forwards.

All four pencils are touching each other.

  • At least seven pencils
  • Modelling clay or adhesive putty, such as Blu-Tack

What to do

Simple four

three pencils form an arrow pointing down, three form an arrow pointing left.

Put a sideways arrow on top of the down arrow.

  1. Put two pencils touching side-by-side on a table.
  2. Put a pencil on top, running across the other two pencils. Then put a fourth pencil next to the third. All four pencils should be touching.

Flat six

  1. Put two pencils on a table in a ‘V’ shape, and make sure they touch.
  2. Put a third pencil in the middle of the ‘V’ so it touches both pencils, making an arrow shape.
  3. On top of the arrow, put two pencils in a sideways ‘V’ shape.
  4. Put the final pencil in the middle of the sideways ‘V’ shape to make a sideways arrow shape.

Martin’s seven solution (using the ends of the pencils)

The ends of three pencils make a triangle.

The ends of three pencils make a triangle. Make a small triangle with the ends of the pencils.

  1. Take three pencils, and measure 15 mm from the blunt end on each pencil
  2. Put the three pencils flat on a table and make a small triangle so that each blunt end touches the 15 mm mark of another pencil. Stick them together with clay and put the triangle to one side.
  3. Put a piece of modelling clay on a table and stick a pencil into the clay so it stands straight up.
  4. Put the little triangle around the vertical pencil, and adjust the size of the triangle so each pencil touches all the others.
  5. Take three more pencils and make a second little triangle just like the first. Hold it together firmly with clay, and then flip it over. If you compare the two small triangles you have, one should now point clockwise, and the other anticlockwise.
  6. Place the new triangle around the vertical pencil, on top of the first triangle. Rotate and adjust it until every pencil touches every other one!

    Seven pencils all touching.

    Notice how the two triangles ‘spin’ in different directions.

What’s happening?

While doing this activity, you might have noticed that you had to put pencils on top of each other to get the big numbers. If you try to get lots of cylinders to touch while they are all flat on a table, you won’t get very far. You can put two pencils side-by-side, or you can get three ends to make a triangle, but there’s no way to get four cylinders to all touch each other in 2D.

If you imagine your cylinders are flexible, then you get a much more interesting 2D question. This is the famous four colour problem, which puzzled mathematicians for over 100 years before being solved with computer assistance.

You could investigate using flexible cylinders in 3D if you want – you could use different colours of wool for each cylinder. It should be possible to have any number of cylinders arranged so they all touch each other – but you might end up with a big knot!

Try another puzzle about getting things to touch

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