Well done Mike.
That’s often the problem when I post two at a time. If one of them gets a bit convoluted, such as the 16 hour clock, the other one gets forgotten about !
Well done Mike.
That’s often the problem when I post two at a time. If one of them gets a bit convoluted, such as the 16 hour clock, the other one gets forgotten about !
Morning IB,
Being somewhat intrigued by your assertion that you had calculated that the ‘precise’ time the hands coincide at 585.140625 …
I have carefully examined a number of possibilities to find where your mistake might have been made.
Of course I can’t be certain how or whereabouts you made the mistake but the most probable scenario seems likely to be along the lines that I indicated a few posts above. Whatever value you got for the number of seconds past 0ne o’clock was almost certainly 585.xxxxxx where .xxxxxx was a fairly small decimal (I got .142857 for example). Quite rightly, you discarded this decimal fraction to leave yourself with 585 as the “nearest whole number of seconds” (which is, of course correct and delivers the correct answer to the Teaser)
However, and this is where I think you made your mistake, you then used a calculator to divide 585 by 64 to find the number of minutes in 585 seconds. The answer is, of course, 9.140625 minutes - note my emphasis on minutes. So the time at which the hands coincide is 9 minutes past One o’clock and a further 0.140625 minutes. I think that you might have mistaken that 9.140625 as the number of seconds beyond the 9 minutes.
Now, 9 minutes, at 64 seconds per minute, accounts for 576 seconds out of the 585. This (obviously) leaves precisely 9 seconds. Of course, the 0.140625 minutes referred to above, also represents these 9 seconds. 0.140625 x 64 = 9 precisely which accounts for the whole of the 585 seconds past one o’clock.
I think, that if you look back carefully at your calculation, you will find that you did (or should have) calculated the time past One o’clock as 585.142857 seconds (ie 585 and one seventh seconds) precisely.
Certainly worth a look.
I know that’s not what I did - I will have to retrace. I started my calcs from 1:00, afterwards adding on the hour. I’m wondering if I might have done something taking 1:00 as 1 in one or other or both of the secs and mins calcs, rather than taking zero.
That’s fair enough IB.
It just struck me as a remarkable, coincidental possibility, given the way the numbers emerged.
Anyway, I’ll post my solution to the basic teaser (ie to the nearest second) this evening and I will follow it up with the “precise” solution shortly thereafter.
Done it! Found the answer to precise overlap that is. I haven’t spent time trying to reproduce what I did yesterday, though it is not starting at the wrong point, so most likely the cause of error is unknown.
The hands will overlap at 585.142857 recurring (that is the 142857 recurring) seconds. So, I get the same figure as you, and indeed it is 1/7th of a sec. which makes me think there is a formula involving:
(hours around clock face -1)/(minutes around clock face -1)
Yes the formula is of that nature.
The hands cross each other 7 times in an 8 hour day (minute hand goes round 8 times whilst the hour hand goes round once).
The crossovers must be evenly spaced because the hands are moving at a constant speed. So the crossovers are at 1/7th hour past 1, 2/7ths hour past 2 etc until finally 7/7ths hour past 7 (namely at 8 o’clock again).
So the exact time is 64x64/7 seconds past 1.
At last ! Well done.
Well done Ravvie.
Elegant solution and nicely described
The picture above illustrates the time that we are seeking
The arithmetic below shows one way of deriving a solution.
Looks like you might have to click on the bottom of the arithmetic section to enlarge it to make it readable - depending on age/eyesight !!!
Time Again
At what precise time will the hands of the 8 hour clock next be diametrically opposite each other ?
Solution comments
These “clock” teasers are no more complicated than “overtaking train” type questions involving relative speeds – using Newton’s ideas and ignoring Einstein’s refinements !!
So, in the diagram below, two trains A and B leave their respective stations X and Y at precisely the same time, travelling towards Z.
Stations X and Y are 10 miles apart. A travels at 60 mph and B travels at 30 mph
How many miles beyond station Y will train A overtake train B ?
Let the required distance = d miles
Distance travelled by A = d + 10
Distance travelled by B = d
Time taken by A to travel = (d + 10) ÷ 60
Time taken by B to travel = d ÷ 30
But these times are the same
Hence: (d + 10) ÷ 60 = d ÷ 30
d + 10 = 2d
d = 10 miles
Train A overtakes Train B 10 miles beyond station Y
And the time taken to reach the overtake position is:
the time Train A takes to travel 20 miles at 60 mph = 20 minutes or
the time Train B takes to travel 10 miles at 30 mph = 20 minutes
The “clock” Teasers can be tackled in the same way.
or…you can use the more direct approach described so elegantly by Ravvie above
I asked my son your original clock question, though in the context of a normal 12 hour clock. He thought about it for a few seconds and said “I think they cross at about 5 minutes and 26 seconds past 1”. I had to resort to a calculator to check and the correct answer was 5 minutes and 27 and 3/11ths seconds (arises from 3600/11). He was actually disappointed to be more than a second out!
I have just looked up at my kitchen clock and noticed the hands were exactly opposite. The time was 21 and 9/11ths minutes to two.
Ravvie,
You, and possibly your son, might like to have a crack at this one that I posted on October 2019. You can find it around about post 346 in this Brain Teaser thread.
Obviously the answer will be a few posts later, probably between post 350 and 400.
let me know how you and your son get on with it.
We need the “precise” answer, ie it will involve a vulgar fraction ! Hope you enjoy it !
Yes, good one. So they first cross after 12/13ths of an hour. Next 1 hour and 11/13ths. 5th crossing at 4 hours and 8/13ths. And it’s the 8/13ths that gives the same answer as you. So the methods work both ways, neat!
I have had a go at some of the teasers in your thread. My favourite is the bricks hitting the lift one. I worked that one out using my A level Physics from many years ago. But I much preferred the answer given of using the lift as the frame of reference - very elegant!
Do you post all the teasers or do you accept ones from others?
Anybody can post a teaser on this thread and you would be more than welcome.
The lift one, that you really liked was posted by Ian and kept a few of us busy for a few days !
Don’t forget guys, I don’t think i’ve seen any responses to that one.
And the Guys “down under” will be waking up in the next hour or so on Sunday morning raring to go …
2/7ths of an hour before two.
The minute hand laps the hour hand every 8/7ths of an hour. So it will have gained half an hour after a further 4/7ths of an hour.
Neat solution Ravvie. Well done !