Tricky Coconut Problem

[Kalkewl8ter]

Five men and a monkey were shipwrecked on a desert island, and they spent the first day gathering coconuts for food. Piled them all up together and then went to sleep for the night.

But when they were all asleep one man woke up, and he thought there might be a row about dividing the coconuts in the morning, so he decided to take his share. So he divided the coconuts into five piles. He had one coconut left over, and he gave that to the monkey, and he hid his pile and put the rest all back together.

By and by the next man woke up and did the same thing. And he had one left over, and he gave it to the monkey. and all five of the men did the same thing, one after the other; each one taking a fifth of the coconuts in the pile when he woke up and each one having one left over for the monkey. And in the morning they divided what coconuts were left and they came out in five equal shares. Of course each one must have known there were coconuts missing; but each one was guilty as the others, so they didn't say anything. How many coconuts were there in the beginning?

[Kalkewl8ter]

Ok, let me rephrase the question:
Determine the lowest possible number of coconuts at the beginning of the night. (The exact value)

[A$$Bandit]

I didn't think monkeys ate coconuts...

[DavidHooper]

Kalkewl8ter, I make it 3121 coconuts in the beginning, and I think this is the smallest possible. The monkey gets 5 and the men get 828
703
603
523
459
each respectively.

Alternative numbers of coconuts could be 6246 or 9371.

[Kalkewl8ter]

David Hooper,

Nice answer, but do you have a solution?

[DavidHooper]

Nice answer, but do you have a solution?

Huh? I thought I just gave a solution?!

How many coconuts were there in the beginning?

vs.

I make it 3121 coconuts in the beginning

[Evil_Giraffe]

I think he means "show your working". Tsk, haven't you just finished your A-levels? Shouldn't that be ingrained still? Don't worry, the alcohol will scrub it from your brain - you don't have to fight it

Anyway, my working (although I didn't follow it through to a conclusion) went something like:
In the morning there must be 5n coconuts where n is integer.
If there are x coconuts left after each person has taken his share then there must be 5x/4 + 1 before that person takes his share.
So:
[1] 25n/4 + 1 is integer
[2] 125n/16 + 5/4 + 1 is integer
[3] 625n/64 + 25/16 + 5/4 + 1 is integer
[4] 3125n/256 + 125/64 + 25/16 + 5/4 + 1 is integer
[5] 15625n/1024 + 625/256 + 125/64 + 25/16 + 5/4 + 1 is integer

This is as far as i got before I went home last night. It's just a case of finding the smallest n that satisfies the above, then working out the answer to eq [5]

[bugzpodder]

yes then just find the common denominator of the eqn[5]

you will end up something like: (15625n+b)/1024+1

where b is an integer constant. then just solve the diaphantine equation
15625n+b=1024x, where n and x are both integers

[alkatran]

lets assume theres 5 at the end.. so.. (5*5)+1 = 26
(26*5)+1 = 131 (im doing this in my head so there could be errors)
25*5 + 5 + 1 = 131... ok thats right....
ok.. so thats equally diveded at end + last of 5 men to get up..

131*5+1=....
100*5 + 30*5 + 1*5 + 1
300+150+5+1
456..... ya thats right...
4/5 men

456*5+1=
400*5 + 50*5 + 6*5 + 1
2000 + 250 + 30 + 1
2281 (now im having fun, god i love numbers)
3/5 men

2281*5 + 1 =
2000*5 + 200*5 + 80*5 + 1*5 + 1
10 000 + 1000 + 400 + 5 + 1
11 406 (jeez.. really starting to go up...)
2/5 men

ah, now for the answer
11 406 * 5 + 1 =
10 000 * 5 + 1000 * 5 + 400*5 + 6*5 + 1
50 000 + 5000 + 2000 + 30 + 1
57 031
1/5 men

the lowest possible number is 57 031, am i right?!
please excuse the lack of parenthese

[DavidHooper]

You need to multiply by 5/4 instead of 5 throughout, because each man takes 1/5 of the current pile and returns the other 4/5. Since you are working backwards, this becomes 5/4.

Hehe, there's a slight slip in the arithmetic in the 2nd paragraph.

Another problem arises in that it's impossible to end up with 5 coconuts remaining.

What's the official answer then?

[alkatran]

ugh.. i cant belive i was doing it as if each man had taken.. well a bit more then he couldve lol....

ok.. so i multiply by 5/4...

26.. man takes 5.. 1 left.. giving... gasp* 20...
so 20 * 5/4 (um.. 1 + 1/4 so.. 1.25...)
20*1.25 + 1 = .... 20 + 5 + 1
ah. so x + (x/4) + 1 = dingding

ok... wait.. why is it impossible to have 5 at the end...? well ill see.. o i c (lol)
ok.. first 5/4 that ends with a integer...
10...no..15no...20, aha

20 + 20/4 + 1
20 + 5 + 1
26
1/5 men

26/4 = ..... uh oh

ok...
up to 40

40 + 10 + 1 = 41
41+41/4=..... (grrrrrr)

time to try 60
60 + 15 + 1 = 76
1/5 men

76/4 = 19 (yaaaaay) (i get the feling its gonna be jsut as bad for the next step )

76 + 19 + 1 = 96
2/5 men

96/4 = 24 (woohoo)
96+24+1=121 (ugh, do they have to take a whole coconut?)

so... im not gonna do all the math... lol... too long..

[Starman]

I whacked a line of formulae into excel, with an extra cell at the end if the final result was an integer.
Copied it down a few times and changed the starting number until I got a TRUE in my end cell.

I agree with DavidHooper, 3121 is the lowest starting number.

[Kalkewl8ter]

David Hooper's answer was correct, but no one has posted a correct SOLUTION thus far.

[Dreamlax]

A solution solves a problem, and the problem is solved, it's 3121. I think what you are asking for is a correct algorithm or method.

[bugzpodder]

nope 3121 is your answer. a solution involves a proof of some kind that arrives at the answer.

[Dreamlax]

Well if that's the case what is wrong with Evil Giraffe's solution? It arrives at the answer... and so does alkatrans. What more could he possibly want!? ARGHRG! Glaven.

Kalkewl8ter, are you looking for a more mathematical solution or just a different one?

[bugzpodder]

I believe she is looking for a complete and correct solution

[SilverSprite]

Yeah i would have to agree with bugz! She i think she wants the easiest solution as well as the solution that makes most sense. Dont go crazy dreamlax. Just keep on looking.

[bugzpodder]

actually all the solutions i've seen here are either incomplete or incorrect

[Kalkewl8ter]

Yes, I am, in fact, looking for a complete and correct solution, as bugz and Silver Sprite have pointed out. David Hooper has given a list of numbers, which happen to be correct, but seem to be randomly generated.

[DavidHooper]

DavidHooper's numbers were not randomly generated.

Here's da proof. Periods for multiplication and errors excepted.

[SilverSprite]

Thats alotta math!

[bugzpodder]

I think i can give a 5 line proof to the question. its basically the same as Hopper's way so no point in posting again.

[Dreamlax]

Originally posted by bugzpodder
actually all the solutions i've seen here are either incomplete or incorrect

Well, just goes to show, if 5 of use were stuck on an island with a monkey, we wouldn't get very far with coconuts!

[Guv]

Back in the prehistoric per-computer era, I went to school with a genius who later became very successful as a lyricist and composer.

Every time some problem was proposed, Steve came up with an intuitive answer in a very short time. This problem was one example.

I started on a brute force solution which probably would have found the correct answer in 20 minutes to an hour. My approach was the following.

Total = 5A + 1
4A = 5B + 1
4B = 5C + 1
4D = 5E + 1
4E = 5F + 1

Make appropriate substitutions and you end up with one equation in two unknowns, which is a linear Diophantine problem. There are known methods (which I have forgotten) for solving such problems.

Steve went into a trance for about 2-3 minutes while I was writing out the above equations and starting to do the second or third substitution, still a long way from the solution, since the first two substitutions are not as messy as the later ones.

At this point Steve came out of his trance with 3121 as the solution. This was the first example of his genius, and I still remembered that number when I read this thread. Here was his solution.

• First it is obvious that adding 3125 (5⁵) to a solution is another solution. To me, this is sort of obvious, not dead cold certain obvious as it was to Steve.
  
• Next, recognize that -4 is a weird solution. Subtract one, getting -5. Then divide into two piles of -4 and -1, allowing you to do it again with the pile of -4.
  
• Finally, add 3125 to -4, giving 3121 as the smallest valid solution.

This was the first of many examples of how the mind of a real genius works. Until I met Steve, I arrogantly thought that nobody could be a lot smarter than I was. Steve made me realize that he and people like Einstein, Bohr, et cetera have minds that function at a level we very intelligent people cannot imagine.

[DavidHooper]

Hi coconut people. If no-one has replied to the proof, does it mean:
a) it's riddled so full of mistakes that no-one can be bothered
b) it's a good proof
c) i didn't explain some point properly
?

[fkauffman]

I found this riddle a couple of years ago.
I could not solve it using my math skills.

I solved it by writing a simple program that tried all possibilities.
And it can be solved.

But if anyone knows the mathematical solution.....

he/she will be my hero for that day (at least)

Fedor

[DavidHooper]

arh! Look for the proof above

[bugzpodder]

Guv that is a sweet sweeet sweet solution from your friend. but i want to stress a point that if you work hard enough, you can get the same solution (like DavidHopper's method). as Einstein said (or maybe Newton i am confused), Genius=1%[something] +99% hard work

[A$$Bandit]

While we're on the subject, he also said "Only two things are infinite: the universe, and human stupidity, and I'm not sure about the former." I kinda love that quote (I hope I got it right...)

[SilverSprite]

Wasnt that newton or somebody of the sort? I'm pretty sure it wasnt einstein.

[SilverSprite]

Albert Einstein? He also said:

"Do not worry about your problems with mathematics, I assure you mine are far greater."

"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. "

"I cannot believe that God would choose to play dice with the universe."

"Great spirits have always found violent opposition from mediocre minds. The latter cannot understand it when a man does not thoughtlessly submit to hereditary prejudices but honestly and courageously uses his intelligence."

"I am enough of an artist to draw freely upon my imagination. Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world."

"The devil has put a penalty on all things we enjoy in life. Either we suffer in health or we suffer in soul or we get fat."
"The pursuit of truth and beauty is a sphere of activity in which we are permitted to remain children all our lives."

"A table, a chair, a bowl of fruit and a violin; what else does a man need to be happy."

"The fear of death is the most unjustified of all fears, for there's no risk of accident for someone who's dead."

"The ideals which have always shone before me and filled me with the joy of living are goodness, beauty, and truth. To make a goal of comfort or happiness has never appealed to me; a system of ethics built on this basis would be sufficient only for a herd of cattle."

"Without deep reflection one knows from daily life that one exists for other people ."

"A hundred times every day I remind myself that my inner and outer life are based on the labors of others ."

"Only a life lived for others is a life worth while ."

"Gravitation can not be held responsible for people falling in love"

"Few are those who see with their own eyes and feel with their own hearts."

lol

[bugzpodder]

Never a Dull Moment!
Einstein Quiz

Albert Einstein wrote this quiz last century. He maintains that 98% of the people in the world cannot solve it.

Facts:

There are five houses in five different colors, in each house lives a person with a different nationality, whom drinks a certain type of beverage, smoke a certain brand of cigar and keep a certain pet. tNo owners have the same pet, smoke the same brand of cigar or drink the same drink.

here's the question: Who owns the fish?

Some criteria:

1. The Brit man lives in a red house.
2. The Swede keeps dogs as pets.
3. The Dane drinks tea.
4. The green house is on the left of the white house.
5. The green house owner drinks coffee.
6. The person who smokes Pall Mall rears birds.
7. the owner of the yellow house smokes Dunhill.
8. The man living in the house right in the center drinks milk.
9. The Norwegian lives in the first house.
10. the man who smokes Blend lives next to the one who keeps cats.
11. The man who keeps horses lives next to the man who smokes Dunhill.
12. The owner who smokes Blue Master drinks beer.
13. The German smokes prince.
14. The Norwegian [lives] next to the blue house.
15. the man who smokes Blend has a neighbour who drinks water

to be fair if you got the answer, PM me.

and i'll tell you if you are right or wrong. if you got the answer you are likely right. Whatever you do don't post the answer up just yet.

[SilverSprite]

Very interesting i'm gonna be stuck doing this now.

[NotLKH]

9. The Norwegian lives in the first house.

Is the first house the first on the right, or first on the left?

[bugzpodder]

i just ordered the house from left to right and assumed left one to be the Norwegian since thats the direction most text goes (left to right). BTW I got another very slick brain teaser called "Quarters in a Circle". Check it out.

[Kalkewl8ter]

Guv, very good, that is the easiest, and most correct solution to the coconut problem.
David Hooper, nice complicated math! I never would have thought of that!

[Kalkewl8ter]

First house: yellow, Norwegian, water, Dunhill, cats
Second house: blue, Dane, tea, Blend, horses
Third house: red, Brit, milk, Pall Mall, birds
Fourth house: green, German, coffee, Prince, fish
Fifth house: white, Swede, beer, Bluemaster, dogs

The German guy has the fish!

[bugzpodder]

okay... okay... (i was going to post the solution until everyone tried it) btw DavidHopper gave the correct answer too

[bugzpodder]

apparently the solution Guv's friend gave is the same as the solution in Martin Gardner's "The 2nd Scientific American Book
of Mathematical Puzzles and Diversions". hmm did the same person gave this solution or is it another example of "great minds think alike?" or highly unlikely but possible, one person took the other's solution and simiply claimed it on their own.

As to the author of that solution, I quote Martin Gardner:

"This solution is sometimes attributed to the University of
Cambridge physicist P. A. M. Dirac, but in reply to my query
Professor Dirac writes that he obtained the solution from
J. H. C. Whitehead, professor of mathematics at Oxford
University (and nephew of the famous philosopher).
Professor Whitehead, answering a similar query, says that
he got it from someone else, and I have not pursued the
matter further."