# Once is Enough

The Strong Men of Titan from Saturn’s largest moon, consider repetition to be a sin. However, they only have 3 sounds that they can make “Kah”, “Brrrr”, and “Umph” (Let’s call them K, B and U for short.) so it is sometimes difficult not to repeat.

What the strong men of Titan mean by repetition is any sequence of sounds that immediately follows the same sequence… so

# UBUBUKUBUKBKUBUKBUKU

is not a sinful utterance because of this:

# UBUBUKUBUKBKUBUKBUKU

but is a sinful utterance because of this:

# UBUBUKUBUKBKUBUKBUKU

Show that the strong men of Titan make 10 sounds in a row without sinning?

### Extensions:

• When the evil Strong Men of Titan want to sing an evil chant, they write X utterances on a wheel and whirl it around and around – singing as it spins. Is it possible that they don’t sin until they have spun the wheel twice?The answer is yes for the following wheel of 4 utterances:

• Find a larger wheel for which this is true?
• Whenever the Strong Men of Titan drive on their orthogonal highways, they never repeat a sequence of turns. Replace “B,U,K” above with “Forward, Left, Right” to get an idea of how they move. After he begins his journey by moving forward, can a Strong Men of Titan ever return home to his mom?

An example of a strong man trying to return home to mom (red dot) without sinning.

• What is the largest N by M paragraph (N,M >1) so that no two rectangles sharing an edge have a repetition? The following 6×6 paragraph is sinful because the red rectangles repeat:

### The Math in This Problem:

This exercise involves mathematical pattern analysis of a long row of alphabets. Using the same line, students are challenged to show that there are ten consecutive sequences without a sequence following an identical one.