Melody

Thanks GingerAle for bringing Hectictar's milesone to our attention.

And to you Hectictar.

We have been greatly honoured to have you so active on our forum.

Thank you for all the wonderful contributions you have made.    

I based this on the fact that the little hand moves 5 minute units in an hour so the little hand is on a minute mark every 12 minutes.

I worked out that ever 12 minutes the hands move apart by a further 11 units.  Of course this is modular so 60units=0units

Then i just did some grunt work and found the pattern. 

There would be a better modular way of doing it but that is how I did it. 

I can give more info if anyone wants it. 

Twice  4:24  and  7:36

An increasing number has its digits increase in size from left to right. For example, 24589 is an increasing number. How many four-digit increasing numbers are there? 

There are 10C4 ways to choose 4 digits that is 210 ways

Now for each of these combiations there is exactly 1 where the digits are in ascenting order.

So that is still 210 ways.

If you discount numbers beginning with 0 then that gets harder.

Lets see... if the first one is 0 how many ways can the others be chosen. 9C3=84 ways

So I think there are 210-84= 126 ways

So there are 126 numbers that meet this criterion.

So 2 minutes past 12 does not count because the hand has gone fractionally past the 12?

That is really cool.

Thanks   XSquaredFactor   

Thanks guest :)

Thanks Ginger,

I certainly like the way your working begins but I haven't worked out this Euler's totient yet.

I'll just take your word for it.  :)

You should not be so critical of our guest.

There is nothing wrong with good intuition and sometimes the thought processes are very hard to explain.

ok thanks, I will think about it some more.

I did it as simultaneous, diophantine equation. :/