Calculus How To

Ducci Sequence

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A Ducci sequence (or N-number game) is a sequence of n-tuples (i.e. a list of integers). Each term in the sequence is formed by finding the absolute difference of neighboring integers [1].

Ducci Sequence Example

Example question: Create a Ducci sequence for the n-tuple (8, 11, 2, 7).

Step 1: Label the n-tuple’s terms a to n:

  • a = 8
  • b = 11
  • c = 2
  • d = 7.

Step 2: Subtract pairs of terms (a and b, b and c, c and d …). The last term is paired with the first time, so in this example d will be subtracted from a:

  • a – b = 8 – 11 = -3
  • b – c = 11 – 2 = 9
  • c – d = 2 – 7 = -5
  • d – a = 7 – 8 = 1.

Step 3: Place the terms from Step 2 into a new n-tuple, ignoring the negative signs (by taking the absolute value):

(3, 9, 5, 1).

Step 4: Connect the new n-tuples from Step 3 with the Ducci sequence so far. We started with (8, 11, 2, 7), so:

(8, 11, 2, 7) → (3, 9, 5, 1).

Step 5: Repeat Steps 1 to 4 until the sequence wither reaches an n-tuple of all zeros or a periodic (repeating loop):

(8, 11, 2, 7) → (3, 9, 5, 1) → (6, 4, 4, 2) → (2, 0, 2, 4) → (2, 2, 2, 1) → (0, 0, 0, 0).

That’s it!

A Graphical Ducci Sequence

The Ducci sequence can also be calculated using squares or circles, although you should start with a large enough shape (perhaps a whole sheet of paper) so that you don’t run out of room.


Example #2: Create a graph for the Ducci sequence (6, 9, 0, 5)

Step 1: Draw a circle and, starting at the top, write the sequence in a clockwise direction.
four numbers game

I’ve highlighted the first term of each iteration in red, so that it will be easier to reconstruct the sequence when we’ve finished.

Step 2: Subtract pairs of terms, starting with the red term and moving in a clockwise direction. Write down the absolute value of each difference:
ducci algorithm

Make sure to highlight the first term, which is 6 – 9 = 3 in this example.

Step 3: Complete the sequence, repeating Step 2 until you get all zeros or have a periodic sequence:
ducci sequence example

References

[1] 4723 Ducci Sequence. Retrieved May 5, 2021 from: https://icpcarchive.ecs.baylor.edu/external/47/4723.pdf

CITE THIS AS:
Stephanie Glen. "Ducci Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/ducci-sequence/
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