Choose any three by three square of dates on a calendar page...
Can you explain how this card trick works?
A combination mechanism for a safe comprises thirty-two tumblers
numbered from one to thirty-two in such a way that the numbers in
each wheel total 132... Could you open the safe?
In this problem we were asked to take any number (less than
$1000$), add the squares of its digits, and then go on repeating
this until a pattern emerges.
You were asked to start at $145$, and you should have got the
sequence $$145 \mapsto42 \mapsto20 \mapsto4 \mapsto16 \mapsto 37
\mapsto58 \mapsto89 \mapsto145$$ and, of course, the sequence then
starts repeating itself.
After trying other starting points you should have been able to
guess that, whatever the starting point, eventually you will always
reach either $1$, when the number is called 'happy' (and then you
will stay at $1$), or the `cycle' given above, when the number is
You were also asked to show that whichever number we start with
(less than $1000$) we always get a number less than $1000$. Any
number less than $1000$ has only one, two or three digits, and each
of the digits is, of course, at most $9$. This means that when we
add the squares of the digits, the largest number we can get is $3
\times(9 \times9)$ which is $243$, and this
is less than $1000$.