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## 'Number Chains' printed from http://nrich.maths.org/

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