If there are n rooms, then there are n cubed coins, with n in each chest. So to avoid bloodshed, n cubed minus n must be divisible by 3, or be zero. This is true for n = 1 to 6, so I am tempted to generalize and say there was no bloodshed.
Well done Eoink and Dozey.
My usual explanation of the relevant part is as follows:
Three integers, n; n-1; n+1 can be re-arranged to n-1; n; n+1
These must be consecutive numbers (integers)
If you have two consecutive numbers, one of them must be even and divisible by 2
With three consecutive numbers, one of them must be divisible by 3
with four consecutive numbers, one of them must be divisible by 4
with five consecutive numbers, one of them mus…
You get the picture
So with three consecutive numbers, one of them must be either zero, three or divisible by three, and hence there is no bloodshed.
For anybody browsing who is wondering how the baron’s treasure teaser is sorted. bear in mind two things;
First, the barber gets one chest of gold, ie “n” coins (not just one coin) - assuming “n” is the number of rooms etc.
Second, the recognition that in three consecutive numbers, one of them has to be a multiple of 3.
I encountered this teaser c.50 years ago, and I still find it interesting.
Eight hour clock
The clock below has been designed for a world which rotates once every 16 hours.
There are 16 hours in their day, 64 minutes in each hour and 64 seconds in each minute.
At present the clock shows a quarter to eight and in a short while the hands will coincide at eight o’clock.
To the nearest second, when will the hands next coincide ? (ie once it has past eight o’clock)
To the nearest second, 9 mins past one, actually coinciling a fraction of a second earlier:
I was probably supposed to do by maths, but it was too essy by eye:
For ease of notation take the face to be marked with 64 even spacings numbered 1-64 clockwise with 64 coinciding with the 8 hour mark.
At precisely 8 secs past one the minute hand will be on 1.00 (division 8) and the hour hand will be 1 division past 1.00 so 9 divisions, and 1 division apart.
One second later the minute hand will be one division further = divn 9, and hour hand will be 1/8th of division further = 9.25 divns, so 0.25 divns apart.
At 1 sec later still at 10 sec past one, the minute hand will be on division 10 and the hour hand will be 0.125 division further, i.e at 9.25 of a division and so 0.75 of a division apart
They coincide between 9 and 10 sec past one, but closer to 9.
I thought as there were three sons, and three categories (rooms, chest and coins) that it would always be divided evenly, so there would be no blood shed. This assume though, that none of the sons hid an even number of coins in his pocket, in which case there would be blood shed.
Hi IB,
Yes, you can do these problems either way, “by eye” or “using arithmetic” (relative speeds basically)
Now, you’ve clearly got the right concept, but your worded description is a tad confusing (to me at least !) … let me explain.
Your first-line answer “To the nearest second, 9 mins past one, actually coinciling a fraction of a second earlier:” isn’t quite correct.
However, your subsequent workings and descriptions reveal that you correctly figured out the time of coincidence is between 9 and 10 minutes past the hour. So not “a fraction of a second earlier (than 9 mins past one)” but obviously a short while after 9 minutes past one.
And the answer in your final line; “They coincide between 9 and 10 sec past one, but closer to 9.” is also frustratingly shy of the mark because the time of coincidence is clearly between 9 and 10 minutes (not seconds as you state) past the hour. But this is probably just an unfortunate typo !!
I’ll post the answer later, just in case you wish to refine you solution “by eye” or you or someone else comes up with the arithmetic
Hi Mike,
I’ve just realised that it’s almost mid-day here in the UK so probably midnight or so “down under”. So when you wake up bright and early, this post will get your Friday morning off to a wonderful start … (well, probably not).
I can see the logic behind your “three categories (rooms … etc”
But just take a couple of low number examples.
Two rooms. Ok ?
So two chests in each room. (ie 4 chests in the treasure house)
Since there are two chests in each room there are two coins in each chest.
Given four chests in the treasure house, this means there are eight coins in total
Eight isn’t divisible by three.
Try three rooms.
So three chests in each room. (ie 9 chests in the treasure house)
Since there are three chests in each room there are three coins in each chest.
Given nine chests in the treasure house, this means there are 27 coins in total
27 is divisible by three
Try four rooms.
So four chests in each room. (ie 16 chests in the treasure house)
Since there are four chests in each room there are four coins in each chest.
Given sixteen chests in the treasure house, this means there are 64 coins in total
64 isn’t divisible by three
With only three shots, we can see that this situation is indeterminate.
It’s only when you give one chest of coins away (eg to the Barber) that the remaining number of coins is reduced to a number that is consistently divisible by three.
I like this teaser, not sure why, but I like it. I got it from the Sunday Times way back in 1969 when we used read it whilst sunbathing on the Persian Gulf with our Heineken or Carlsberg and BBQ steak sandwiches.
Hi Don,
Drat! That’s what comes of trying to type when one’s eyes are closing late at night trying to get it in before anyone else!
To the nearest second, 9 minutes and 9 seconds past one o’clock (or if you prefer 585 seconds past 1). I see I even missed the seconds out in first part of answer!
If rotating smoothly without significant stepping the exact time of the hands coinciding is 0.140625 seconds later.
You got it ! well done.
I’ll give you or others a few hours to explain the calculation before posting my arithmetic.
BTW. I don’t agree with the “0.140625 seconds later”. It’s worth remembering that the 585 seconds that you mentioned isn’t exact. It should be 585(and one seventh) seconds.
My “exact” answer would be 9 minutes and 9(and one seventh) seconds past one o’clock. And as decimals that is a recurring number and not quite the same as 0.140625
Btw IB, you did well to get the clock sorted out.
That last little bit about one seventh of a second is only so that if you use this teaser with your friends or family, you avoid any difficult challenges.
Cheers
Don
In my case calculated (precisely) as 585.140625 Secs past, and therefore is correct to say 585 to the nearest second.
0.140625 is 9/64 - nearly 1/7th but not quite,
However, I don’t see a solution with 7ths, even if only aporox!
But maybe I’m latching onto something unintended in your answer…
Hi IB,
First, the teaser asked for the answer “to the nearest second” and for sure you have got the answer correct at 9 mins and 9 secs past one o’clock. Full marks and a well-deserved “Well Done!”
You then volunteered the extra information about the “precise” time the hands co-inside. Now this is where we clearly have a difference. You have a figure 585.140625 secs. It’s not part of the Teaser as such, but it is definitely worth exploring to see if we can get it right.
Now, I arrived at a calculated figure 585.142857142857. (I actually arrived at 585 and one seventh because I did it by hand and left it as a vulgar fraction, but subsequently continued the long division and wrote out the recurring decimal).
The final line of my calculations generates X = 512 x 8/7 = 585 and one seventh seconds or, as a decimal = 585.142857 which, because the Teaser only requires the answer “to the nearest second” means 585 secs.
Now, after a bit of “playing about”, I have stumbled upon a computer calculation that does generate the number 9.140625 and which to my initial horror, looked convincing like 9.140625 seconds. If you take 585 (ie the number of seconds without the decimal part) and divide it by 64, you get … 9.140625
I can visualise the possibility of someone taking 585 and using a calculator to divide it by 64 to find the number of whole minutes. Sure, the answer is 9, but the remainder (0.140625) is meaningless because 585 omits the 0.142857 part of the true time.
Perhaps a quick check of your calcs will show that you did indeed have 585.142857 as the precise number of seconds ? and simply slipped up with the conversion into minutes and seconds ?
Now, that does sound like fun indeed!
Yes. It sure was !
No, I got 585.140625 as the precise number of seconds.
But I have been trying to prove it in as simple a way I can, and there appears to be an anomaly. Not sure if that means I’ve cocked up the proof, or the original calc (which you think is the case). I’ve spent too much time so will have to leave for now. I’m beginning to wonder if I slipped a line in my original calc…
No problem IB.
If nobody else comes up with the “precise” number of seconds, either to support your figure, mine or yet some other figure, I’ll post my calculation tomorrow.
We might then be able to determine when the precise coincidence of hands occurs.
I’m intrigued.
Meanwhile, about 14 posts above is an “easy filler” that is lying untouched …
30? There are five even numbers that are followed by an odd number (4, 4, 8, 6 & 2). 5 x 6 = 30.