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Digit-Sum Spirals

What is a spiral? Can you draw a spiral on a squared grid?

Start by finding some digit-sums:

24 > 2+4 = 6

47 > 4+7 = 11 > 1+1 = 2

149 > 1+4+9 = 14 > 1+4 = 5

So the digit sum of 149 is 5


Now create a digit-sum spiral for multiples of 3:

  1. Write out multiples of 3 in a column ( up to 15 x 3?)Write the digit-sum alongside each number. (3,6,9,3,6,9,3,6,9,3......)
  2. Discuss any patterns that can be seen in the sequence of digit sums.
  3. Start to draw the spiral (0.25 cm squared paper is recommended) :
  4. Starting point is at the top of the page, in the centre.
  5. First line : RIGHT 3 squares
  6. Turn clockwise 90 degrees, then DOWN 6 squares
  7. Turn clockwise 90 degrees, then LEFT 9 squares.....
  8. Turn clockwise 90 degrees, then UP 3 squares...
  9. Continue turning and using the digit sums until the starting point is reached: the 'spiral' is now complete!

It is perfectly OK to draw over part of a line that has already been drawn:


Digit-sum spiral for multiples of 3


Now try using multiples of other numbers:

1,2,6,7,8,9 are straightforward; 4 and 5 are slightly trickier.

You will need to experiment to find a good starting point so that the spiral doesn't go off the paper.

Compare results:

Do any numbers have similar digit-sum spirals?

What about multiples of 10? 11? 12?.....Can you make any generalized statements?

See some of the spirals here.


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