I chose a certain sequencing rule, and I told my friend no matter what number he named, it would eventually end up at repeating 4s.

At first he chose 5, which ended in 4, 4, 4... Then he tried 18, which met the same fate. Even numbers like -42 and Pi ended in 4, 4, 4...

Still trying to outdo me, he started to get really creative. He tried 63424563562324 ... same. He even tried a googolplex and Avogadro's number. But alas all the numbers he tried ended up in 4,4,4. Finally he conceded.

What's the sequencing rule I use? Should he have conceded? Can you find any starting terms which don't end in 4, 4, 4...?

(In reply to Let me spell it out for you (Solution) by Jyqm)

"Four is the only number which contains the same number of letters as it denotes, so any sequence by this rule would eventually reduce to repeating fours."

This is not enough: you must also show that there are no cycles (loops). For example, in Spanish you could end in an all 5 (CINCO) sequence, but you could also loop between 4 (CUATRO) and 6 (SEIS).

Finally, you should also show that any number eventually produces a descending sequence: i.e., that a sequence such as 1 (ONE) => 3 (THREE) => 5 (FIVE) eventually goes down.

Posted by Federico Kereki on 2007-07-31 16:28:13