Home > Problem of the Week > Archive List > Detail

<< Prev 1/4/2009 Next >>

Caterpillar Numbers--Type B

Pick a natural number n<100

Rewrite the number n = 10T + U, where U is the unit's digit and T is the ten's digit

The next number segment in the caterpillar number is 4U+T

Repeat using your new result as n

The caterpillar number ends when its "head" equals its "tail"

The chain of numbers becomes a caterpillar number of Type B.

For example, two caterpillar numbers of this type are 12-9-36-27-30-3-12-9 and 6-24-18-33-15-21-6.

Some Questions To Explore:

  • Are all such caterpillar numbers finite in length?
  • What is the shortest caterpillar number you can find? Longest?
  • Is there a pattern formed by caterpillar numbers of the same length?
  • Is there an "even-odd-even...odd..." caterpillar number?
  • What is special about the number 13?
  • What happens if you try initial n-values greater than 100?
  • What are caterpillar numbers of Type A....see the previous problem in the Archive?
Extension: Expand your exploration by trying 2U+T, 3U+T, 5U+T, etc....

Some Trivia: The diagram is part of a fabric called "Caterpillar Numbers," available on-line.


Source: J.M. (Bellingham)...who loves to explore number patterns.

Hint: Again, you just need to play with this pattern. No real "answer" is the goal....rather the goal is the discovery process itself.


Solution Commentary: Send me your discoveries....I am prepared to include them as part of this commentary.

I am especially interested in proofs or arguments relative to the special nature of 13 and the generalization to n>100 or for the rule pU+t for p a natutal number.