Twenty numbers in the range from 1 to 100 are happy, and they
may be presented in a tree diagram showing the chain of
numberswhich takes them to 1. Know the time It takes only a few
seconds' reflection to appreciate that once every hour the
minute hand and the hour hand must point in precisely opposite
directions. But how often in a day will each hand be pointing
exactly at a minute division at the same time as the hands are
precisely opposite each other? When the surface of a lake or
pond is undisturbed, it behaves like a plane horizontal mirror.
The laws of reflection of light (angle of incidence equals angle
of reflection) operate and only the light (from a point source
on the opposite bank) reflected from a particular point of the
surface can enter our eyes. This ensure that we see a clear
image of the light source. However, when the surface becomes
wavy due to the action of the wind, there are multiple points on
it that are so inclined relative to us that they canall reflect
the light into our eyes, and