Watches

There is an old joke for those who still have analogue watches which consist of hands for second, minute and hour.

What did the second hand say to the hour hand as it went past? See you in a minute.

That begs a question. Precisely how long does it take the second hand to catch up with the hour hand?

The calculation is not as simple as it appears

Rpm is revolutions per minute

Note there are		1440	minutes in an hour
		60	seconds in a minute
	angular velocity
	Rpm
hour hand	0.000694444		= 1 / 1440	A
second hand	1			B
bring hour hand to halt by applying negative velocity,
means relative to hour hand                            angular velocity
	Rpm
hour hand	0			C	A - A
second hand	0.999305556			D	B - A
Thus time taken for hour hand and second hand to be aligned is
	1.0006949	minutes		=1/D
	60.041696	seconds

In other words see you in a minute is about right!

Reunions

Mathematics Blog

This is the first in an occasional blog where I try and recall the mathematics which I have learnt and apply it in situations.

…then eventually, I will try and explain in a longer manner where it could go and where applicable why I came up with the problem.

Reunion

A reunion took place with 8 persons. If each person hugged everyone else once, how many hugs (of two persons) took place?

The answer is of course ⁸C₂ -= 8!/(6! 2!) = 8 × 7 / (2 × 1) = 28

However if the eight consist of three married couples and two single people, and the spouses did not hug each other, the number of hugs becomes 28 – 3 = 25

In general, if there are n persons including r couples implies    ⁿC₂ – r    hugs took place.

The longer discussion

I thought about this when wondering when 8 of us met up how many hugs there would be. I of course calcualted 28 then I wondered why there was not 28 hugs and then I realised we arrived in three groups…one on his own, two couples together and I and another couple. This was followed by the fact the couples did not hug… So you may be interested to work out how many hugs I needed to do and see once we all met…assuming those who arrived did not hug those they arrived with! It might also be worth wondering how long it took to hug.

In general Discrete Mathematics has plenty of applications. The mathematics would of course refer to pairs etc or perhaps unordered pairs. Also threesomes or triples and other n-tuples could be considered.