Number Chains

We can express any whole number $n$ as $n=10a + b$ where $b$ is the remainder obtained when $n$ is divided by 10. The chaino sequence is defined by the mapping $10a + b \to 2a + 3b$. For example $1357 \to 291 \to 61 \to 15 \to 17 \to 23 ...$.

Investigate this sequence of numbers using different starting points.

You will find that 14 is a fixed point and there are periodic cycles like the cycle of length 6 : $18\to 26 \to 22 \to 10 \to 2 \to 6 \to 18$.

Prove that, for all starting points, the numbers in the sequence quickly reduce to numbers less than 45 and do not later increase above 45. How many periodic cycles are there and how many fixed points?

Explain why the numbers in a cycle are either all even or all odd.

You could investigate the sequence using a calculator or a spreadsheet. Alternatively the following Logo program will generate the sequence. It uses the fact that, if $n=10a + b$, then $a$ is the integer part of ${n\over 10}$. Try loading the program and typing chaino 1357 . You will need to click on HALT when you want to stop the program.

to chaino :n
make "n 2*(int :n/10) + 3*(:n - 10*int :n/10)
print :n
wait 120
chaino :n
end