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标题: Hard problem this time [打印本页]

作者: z-score 时间: 2009-10-2 11:20:06 标题: Hard problem this time

本帖最后由 z-score 于 2009-10-2 12:16 编辑

"Five men were stranded onan island. They went around picking up
coconuts for years. One day,they saw a ship coming. They had a radio
so they sent a message to comeand pick them up. The ship said "yes,
tomorrow morning!"

The five men went to sleep butthe first man woke up and thought "I
don't trust my buddies,"so he took 1/5 of the pile of coconuts.

Then a monkey came down andtook 1 coconut.

The second man woke up anddidn't trust his buddies either, so he took
1/5 of the remaining pile ofcoconuts.
The monkey came down again and
took 1 coconut.

During the rest of the night,the third, fourth, and fifth men did the
same and the monkey took 1coconut after each man.

In the morning, the five mentried to divide the remaining pile of
coconuts into five equalportions but had one left over, which they
gave to the monkey.
What is the least number of coconuts you could have in the original pile?

[fly]Please don't Google, otherwise there is no challenge![/fly]

作者: love_3_month 时间: 2009-10-2 11:56:29

12495? seems too big (and ridiculous?) i must have missed something.......

作者: victordd 时间: 2009-10-2 11:59:30

数学没学好。只会列方程。。不会算。。。

作者: ceerose 时间: 2009-10-2 12:01:49

不难啊，留给其他人来解吧

作者: ceerose 时间: 2009-10-2 12:05:05

2# love_3_month

应该是这个，好勤劳的五个人啊，而且他们也很能搬，想想第一个醒来的人要搬走多少个椰子，好体力

作者: ceerose 时间: 2009-10-2 12:07:59

3# victordd

列完表面的，还要列隐含的。。

作者: z-score 时间: 2009-10-2 12:16:30

12495? seems too big (and ridiculous?) i must have missed something.......
love_3_month 发表于 2009-10-2 11:56

Sorry, I should change the question to, what is the least number of coconuts you could start with.

作者: z-score 时间: 2009-10-2 12:17:37

不难啊，留给其他人来解吧
ceerose 发表于 2009-10-2 12:01

Question changed, sorry, didn't have the right one before.

作者: 不是排骨不熬汤 时间: 2009-10-2 13:00:06

关键在最后，可以减一后，除5 得最小整数。

12495 可以得出整数818. - 有点大。。

作者: z-score 时间: 2009-10-2 13:18:58

yeah still too big. keep trying though.

作者: love_3_month 时间: 2009-10-2 13:51:16

here is my silly way (by excel, just put formula in A1, - F1, then copy down. )

the last pile should be 6, 11, 16, etc, (=x, put in A1 = 6, a2 = A1+5, copy down)
the previous one should be (x+1)*1.25 which is an integer, so x is 11/31/51/71/91 etc (B1 = (A1+1)*1.25)
the previous one should be ((x+1)*1.25+1)*1.25, which is an integer, so x is 11/91/171/251 etc (C1=(B1+1)*1.25)
the previous one should be (((x+1)*1.25+1)*1.25+1)*1.25, which is an integer, x is 251/571/891/1211 etc
the previous one should be ((((x+1)*1.25+1)*1.25+1)*1.25+1)*1.25, which is an integer, x is 251/1531/2811/4091 etc
the previous one should be (((((x+1)*1.25+1)*1.25+1)*1.25+1)*1.25+1)*1.25, which is an integer, x is 4091/9211/14331/19451 etc

when x = 4091, the original pile =12495

there must be something wrong, but i do know where.........

作者: saintjohn 时间: 2009-10-2 14:18:24

本帖最后由 saintjohn 于 2009-10-2 13:22 编辑

just work backwards:
for the original pile to have the least number of coconuts, the coconut at last day must be 6 (1 each and monkey gets one)
then before the 5th man took it, there were (6 + 1) x 5 = 35 coconuts
then before the 4th man took it, there were (35 +1) x 5 = 180 coconuts
then before the 3rd man took it, there were (180+1) x 5 = 905 coconuts
then before the 2nd man took it, there were (905+1) x 5 = 4530 coconuts
then originally (before the 1st man took it), there were (4530+1) x 5 = 22655 coconuts

作者: love_3_month 时间: 2009-10-2 14:21:45

one man took 1/5, not left 1/5

your solution is like they took 4/5........

作者: saintjohn 时间: 2009-10-2 14:23:57

13# love_3_month

oh true, stupid me

作者: saintjohn 时间: 2009-10-2 14:31:40

i guess in math term it is:

N original = ((((((5X +1)1.25 + 1)1.25 +1)1.25 + 1 )1.25 + 1)1.25 +1)1.25 +1

find smallest X to make N a positive integer.

is it?

作者: ceerose 时间: 2009-10-2 14:40:02

15# saintjohn

you need a few more formulas than that

作者: love_3_month 时间: 2009-10-2 15:02:21

i think the formula should be

N original =+((((((X*5+2)*1.25+1)*1.25+1)*1.25+1)*1.25+1)*1.25) and make N integer

i still get the same result. x = 818 , N = 12495

作者: z-score 时间: 2009-10-2 15:21:32

just work backwards:
for the original pile to have the least number of coconuts, the coconut at last day must be 6 (1 each and monkey gets one)
then before the 5th man took it, there were (6 + 1) x ...
saintjohn 发表于 2009-10-2 14:18

The answer is around 3000.

作者: z-score 时间: 2009-10-2 15:22:05

i think the formula should be

N original =+((((((X*5+2)*1.25+1)*1.25+1)*1.25+1)*1.25+1)*1.25) and make N integer

i still get the same result. x = 818 , N = 12495
love_3_month 发表于 2009-10-2 15:02

You have a solution, but not the solution with the least possible number of coconuts.

作者: 夜.冥狼 时间: 2009-10-2 15:34:29

Well.......

3120 ??

total ??

作者: saintjohn 时间: 2009-10-2 15:46:34

Well.......

3120 ??

total ??
夜.冥狼发表于 2009-10-2 14:34

how did u work it out, bro?

作者: love_3_month 时间: 2009-10-2 15:48:20

if you use 3120, the last pile in the morning will be 1019, which cannot be didived by 5 and left 1.

作者: 夜.冥狼 时间: 2009-10-2 15:49:45

21# saintjohn

I didn't ....

I got stuck exactly at where you get stuck... I.e. Know there are some other constraints besides the numbers, but could not put them into formula to solve them

3120 is an cheated answer from Excel.......

But even 3120 doesn't look right...

作者: 夜.冥狼 时间: 2009-10-2 16:24:34

if you use 3120, the last pile in the morning will be 1019, which cannot be didived by 5 and left 1.
love_3_month 发表于 2009-10-2 15:48

Exactly...

But 3120 is the closest in the 2500-3500 range....

(Yes, I test all these answer using excel.......)

作者: 夜.冥狼 时间: 2009-10-2 16:28:31

22# love_3_month I am now quite convinced that 12495 is the right (smallest) answer. Because I think there should be some kind of characteristic for the answer, and 12495 seems the lowest possible number that fit the characteristics.........

Or maybe just like ATM said..... Our logic is on the wrong track to find the smallest number.........

LOL>>>>

作者: z-score 时间: 2009-10-2 16:28:58

Exactly...

But 3120 is the closest in the 2500-3500 range....

(Yes, I test all these answer using excel.......)
夜.冥狼发表于 2009-10-2 16:24

The answer has to end with either 1 or 6. Because when divide by 5, 1 is the remainder.

作者: love_3_month 时间: 2009-10-2 16:40:20

22# love_3_month I am now quite convinced that 12495 is the right (smallest) answer. Because I think there should be some kind of characteristic for the answer, and 12495 seems the lowest possible n ...
夜.冥狼发表于 2009-10-2 16:28

well i checked every number starting from 1. (yes i am cheating because i only know excel)

12495 is the first number meeting all criteria.

and from google i found the below thing, it got the same answer, but from another way. (seems much more professional than my Excel)

ATM does not think the below solution though. Anyone can read this through and check again?

but I am now more confident my answer is right kaka.......

http://mathforum.org/library/drmath/view/56769.html

This problem, or variants of it, have been around for a long time.
This is how you solve it.

Let a be the number of coconuts to start with. After the first man and
the monkey took their coconuts, the number left was b = (4/5)*a - 1.
After the second man and the monkey took their coconuts, the number
left was c = (4/5)*b - 1. The third man left d = (4/5)*c - 1 coconuts.
The fourth man left e = (4/5)*d - 1 coconuts.  The fifth man left
f = (4/5)*e - 1 coconuts.  At the end, f = 5*g + 1, where g is the
number of coconuts each man got in the morning.

Using the last equation, substitute in the previous one to get

         5*g + 1 = (4/5)*e - 1,
      5*(5*g+2) = 4*e,
      25*g + 10 = 4*e.

Now substitute that in the equation relating d and e:

      25*g + 10 = 4*[(4/5)*d - 1],
   5*(25*g+14) = 16*d,
      125*g + 70 = 16*d.

Now substitute that in the equation relating c and d:

      125*g + 70 = 16*[(4/5)*c - 1],
   5*(125*g+86) = 64*c,
   625*g + 430 = 64*c.

Now substitute that in the equation relating b and c:

   625*g + 430 = 64*[(4/5)*b - 1],
   5*(625*g+494) = 256*b,
   3125*g + 2470 = 256*b.

Now substitute that in the equation relating a and b:

   3125*g + 2470 = 256*[(4/5)*a - 1],
5*(3125*g+2726) = 1024*a,
15625*g + 13630 = 1024*a,
  1024*a - 15625*g = 13630.

Any positive whole numbers a and g which are a solution of this
equation will give you a solution to your original problem.  The
problem has been reduced to finding the value of a.

Notice that since 5 divides 15625 and 13630, but not 1024, that 5 must
divide a, so a = 5*h.  Then

1024*5*h - 15625*g = 13630,

or, dividing by 5 througout,

   1024*h - 3125*g = 2726.

Similarly, since 2 divides 1024 and 2726, but not 3125, 2 must divide
g, so g = 2*i, and

1024*h - 3125*2*i = 2726,
   512*h - 3125*i = 1363.

Now we can tell that i must be odd, so i = 2*j + 1, and

512*h - 3125*(2*j+1) = 1363,
   512*h - 3125*2*j = 1363 + 3125 = 4488,
   256*h - 3125*j = 2244.

Next, since 4 divides 256 and 2244, but 2 doesn't divide 3125, 4 must
divide j, so j = 4*k, and

   256*h - 3125*4*k = 2244,
      64*h - 3125*k = 561.

Similarly, k must be odd, so k = 2*m + 1, and

32*h - 3125*m = 1843.

Similarly, m must be odd, so m = 2*n + 1, and

16*h - 3125*n = 2484.

Now n must be divisible by 4, so n = 4*p, and

4*h - 3125*p = 621.

Again, p must be odd, so p = 2*q + 1, and

2*h - 3125*q = 1873.

Further, q must be odd, so q = 2*r+1, and

   h - 3125*r = 2499.

One solution to this is r = 0, h = 2499.  There are other, larger
ones, such as r = 1, h = 5624, and r = 100, h = 314999, but we will
stick with the smallest one.  The general solution is h = 2499 + 3125*
r, r any nonnegative whole number.

Backtracking, we get q = 1, p = 3, n = 12, m = 25, k = 51, j = 204,
i = 409, h = 2499, g = 818, f = 4091, e = 5115, d = 6395, c = 7995,
b = 9995, and a = 12495.  Sure enough,

1024*12495 - 15625*818 = 13630.

Does this work?  Let's check.

Starting with 12495:
The first man took 2499 coconuts, and the monkey took 1.
  This left 12495 - 2500 = 9995 coconuts.
The second man took 1999 coconuts, and the monkey took 1.
  This left 9995 - 2000 = 7995 coconuts.
The third man took 1599 coconuts, and the monkey took 1.
  This left 7995 - 1600 = 6395 coconuts.
The fourth man took 1279 coconuts and the monkey took 1.
  This left 6395 - 1280 = 5115 coconuts.
The fifth man took 1023 coconuts and the monkey took 1
  This left 5115 - 1024 = 4091 coconuts.
In the morning, each man got 818 coconuts and the monkey 1 more.

It checks!  Hooray!

If we backtrack using h = 2499 + 3125*r, we will find that the general
answer is a = 12495 + 15625*r coconuts, for any nonnegative whole
number r.

The men must have done a whole lot of counting in the middle of the
night!

作者: hitye 时间: 2009-10-2 16:41:46

lsd have empty

作者: 走两步 时间: 2009-10-2 18:59:15

12495, matlab 求证结果

作者: 快乐的猴子 时间: 2009-10-2 19:09:28

possible_value = ((((((X*5+2)*1.25+1)*1.25+1)*1.25+1)*1.25+1)*1.25) ;

output :
x=818 possible_value=12495
x=1842 possible_value=28120
x=2866 possible_value=43745
x=3890 possible_value=59370
x=4914 possible_value=74995
x=5938 possible_value=90620
x=6962 possible_value=106245
x=7986 possible_value=121870
x=9010 possible_value=137495
x=10034 possible_value=153120
x=11058 possible_value=168745
x=12082 possible_value=184370
x=13106 possible_value=199995
x=14130 possible_value=215620
x=15154 possible_value=231245
x=16178 possible_value=246870
x=17202 possible_value=262495
x=18226 possible_value=278120
x=19250 possible_value=293745
x=20274 possible_value=309370
x=21298 possible_value=324995
x=22322 possible_value=340620
x=23346 possible_value=356245
x=24370 possible_value=371870
x=25394 possible_value=387495
x=26418 possible_value=403120
x=27442 possible_value=418745
x=28466 possible_value=434370
x=29490 possible_value=449995
x=30514 possible_value=465620
x=31538 possible_value=481245
x=32562 possible_value=496870
x=33586 possible_value=512495
x=34610 possible_value=528120
x=35634 possible_value=543745
x=36658 possible_value=559370
x=37682 possible_value=574995
x=38706 possible_value=590620
x=39730 possible_value=606245
x=40754 possible_value=621870
x=41778 possible_value=637495
x=42802 possible_value=653120
x=43826 possible_value=668745
x=44850 possible_value=684370
x=45874 possible_value=699995
x=46898 possible_value=715620
x=47922 possible_value=731245
x=48946 possible_value=746870
x=49970 possible_value=762495
x=50994 possible_value=778120
x=52018 possible_value=793745
x=53042 possible_value=809370
x=54066 possible_value=824995
x=55090 possible_value=840620
x=56114 possible_value=856245
x=57138 possible_value=871870
x=58162 possible_value=887495
x=59186 possible_value=903120
x=60210 possible_value=918745
x=61234 possible_value=934370
x=62258 possible_value=949995
x=63282 possible_value=965620
x=64306 possible_value=981245
x=65330 possible_value=996870
x=66354 possible_value=1012495
x=67378 possible_value=1028120
x=68402 possible_value=1043745
x=69426 possible_value=1059370
x=70450 possible_value=1074995
x=71474 possible_value=1090620
x=72498 possible_value=1106245
x=73522 possible_value=1121870
x=74546 possible_value=1137495
x=75570 possible_value=1153120
x=76594 possible_value=1168745
x=77618 possible_value=1184370
x=78642 possible_value=1199995
x=79666 possible_value=1215620
x=80690 possible_value=1231245
x=81714 possible_value=1246870
x=82738 possible_value=1262495
x=83762 possible_value=1278120
x=84786 possible_value=1293745
x=85810 possible_value=1309370
x=86834 possible_value=1324995
x=87858 possible_value=1340620
x=88882 possible_value=1356245
x=89906 possible_value=1371870
x=90930 possible_value=1387495
x=91954 possible_value=1403120
x=92978 possible_value=1418745
x=94002 possible_value=1434370
x=95026 possible_value=1449995
x=96050 possible_value=1465620
x=97074 possible_value=1481245
x=98098 possible_value=1496870
x=99122 possible_value=1512495
x=100146 possible_value=1528120
x=101170 possible_value=1543745
x=102194 possible_value=1559370
x=103218 possible_value=1574995
x=104242 possible_value=1590620
x=105266 possible_value=1606245
x=106290 possible_value=1621870
x=107314 possible_value=1637495
x=108338 possible_value=1653120
x=109362 possible_value=1668745
x=110386 possible_value=1684370
x=111410 possible_value=1699995
x=112434 possible_value=1715620
x=113458 possible_value=1731245
x=114482 possible_value=1746870
x=115506 possible_value=1762495
x=116530 possible_value=1778120
x=117554 possible_value=1793745
x=118578 possible_value=1809370
x=119602 possible_value=1824995
x=120626 possible_value=1840620
x=121650 possible_value=1856245
x=122674 possible_value=1871870
x=123698 possible_value=1887495
x=124722 possible_value=1903120
x=125746 possible_value=1918745
x=126770 possible_value=1934370
x=127794 possible_value=1949995
x=128818 possible_value=1965620
x=129842 possible_value=1981245
x=130866 possible_value=1996870
x=131890 possible_value=2012495
x=132914 possible_value=2028120
x=133938 possible_value=2043745
x=134962 possible_value=2059370
x=135986 possible_value=2074995
x=137010 possible_value=2090620
x=138034 possible_value=2106245
x=139058 possible_value=2121870
x=140082 possible_value=2137495
x=141106 possible_value=2153120
x=142130 possible_value=2168745
x=143154 possible_value=2184370
x=144178 possible_value=2199995
x=145202 possible_value=2215620
x=146226 possible_value=2231245
x=147250 possible_value=2246870
x=148274 possible_value=2262495
x=149298 possible_value=2278120
x=150322 possible_value=2293745
x=151346 possible_value=2309370
x=152370 possible_value=2324995
x=153394 possible_value=2340620
x=154418 possible_value=2356245
x=155442 possible_value=2371870
x=156466 possible_value=2387495
x=157490 possible_value=2403120
x=158514 possible_value=2418745
x=159538 possible_value=2434370
x=160562 possible_value=2449995
x=161586 possible_value=2465620
x=162610 possible_value=2481245
x=163634 possible_value=2496870
x=164658 possible_value=2512495
x=165682 possible_value=2528120
x=166706 possible_value=2543745
x=167730 possible_value=2559370
x=168754 possible_value=2574995
x=169778 possible_value=2590620
x=170802 possible_value=2606245
x=171826 possible_value=2621870
x=172850 possible_value=2637495
x=173874 possible_value=2653120
x=174898 possible_value=2668745
x=175922 possible_value=2684370
x=176946 possible_value=2699995
x=177970 possible_value=2715620
x=178994 possible_value=2731245
x=180018 possible_value=2746870
x=181042 possible_value=2762495
x=182066 possible_value=2778120
x=183090 possible_value=2793745
x=184114 possible_value=2809370
x=185138 possible_value=2824995
x=186162 possible_value=2840620
x=187186 possible_value=2856245
x=188210 possible_value=2871870
x=189234 possible_value=2887495
x=190258 possible_value=2903120
x=191282 possible_value=2918745
x=192306 possible_value=2934370
x=193330 possible_value=2949995
x=194354 possible_value=2965620
x=195378 possible_value=2981245
x=196402 possible_value=2996870
x=197426 possible_value=3012495
x=198450 possible_value=3028120
x=199474 possible_value=3043745
x=200498 possible_value=3059370
x=201522 possible_value=3074995
x=202546 possible_value=3090620
x=203570 possible_value=3106245
x=204594 possible_value=3121870
x=205618 possible_value=3137495
x=206642 possible_value=3153120
x=207666 possible_value=3168745
x=208690 possible_value=3184370
x=209714 possible_value=3199995
x=210738 possible_value=3215620
x=211762 possible_value=3231245
x=212786 possible_value=3246870
x=213810 possible_value=3262495
x=214834 possible_value=3278120
x=215858 possible_value=3293745
x=216882 possible_value=3309370
x=217906 possible_value=3324995
x=218930 possible_value=3340620
x=219954 possible_value=3356245
x=220978 possible_value=3371870
x=222002 possible_value=3387495
x=223026 possible_value=3403120
x=224050 possible_value=3418745
x=225074 possible_value=3434370
x=226098 possible_value=3449995
x=227122 possible_value=3465620
x=228146 possible_value=3481245
x=229170 possible_value=3496870
x=230194 possible_value=3512495
x=231218 possible_value=3528120
x=232242 possible_value=3543745
x=233266 possible_value=3559370
x=234290 possible_value=3574995
x=235314 possible_value=3590620
x=236338 possible_value=3606245
x=237362 possible_value=3621870
x=238386 possible_value=3637495
x=239410 possible_value=3653120
x=240434 possible_value=3668745
x=241458 possible_value=3684370
x=242482 possible_value=3699995
x=243506 possible_value=3715620
x=244530 possible_value=3731245
x=245554 possible_value=3746870
x=246578 possible_value=3762495
x=247602 possible_value=3778120
x=248626 possible_value=3793745
x=249650 possible_value=3809370
x=250674 possible_value=3824995
x=251698 possible_value=3840620
x=252722 possible_value=3856245
x=253746 possible_value=3871870
x=254770 possible_value=3887495
x=255794 possible_value=3903120
x=256818 possible_value=3918745
x=257842 possible_value=3934370
x=258866 possible_value=3949995
x=259890 possible_value=3965620
x=260914 possible_value=3981245
x=261938 possible_value=3996870
x=262962 possible_value=4012495
x=263986 possible_value=4028120
x=265010 possible_value=4043745
x=266034 possible_value=4059370
x=267058 possible_value=4074995
x=268082 possible_value=4090620
x=269106 possible_value=4106245
x=270130 possible_value=4121870
x=271154 possible_value=4137495
x=272178 possible_value=4153120
x=273202 possible_value=4168745
x=274226 possible_value=4184370
x=275250 possible_value=4199995
x=276274 possible_value=4215620
x=277298 possible_value=4231245
x=278322 possible_value=4246870
x=279346 possible_value=4262495
x=280370 possible_value=4278120
x=281394 possible_value=4293745
x=282418 possible_value=4309370
x=283442 possible_value=4324995
x=284466 possible_value=4340620
x=285490 possible_value=4356245
x=286514 possible_value=4371870
x=287538 possible_value=4387495
x=288562 possible_value=4403120
x=289586 possible_value=4418745
x=290610 possible_value=4434370
x=291634 possible_value=4449995
x=292658 possible_value=4465620
x=293682 possible_value=4481245
x=294706 possible_value=4496870
x=295730 possible_value=4512495
x=296754 possible_value=4528120
x=297778 possible_value=4543745
x=298802 possible_value=4559370
x=299826 possible_value=4574995
x=300850 possible_value=4590620
x=301874 possible_value=4606245
x=302898 possible_value=4621870
x=303922 possible_value=4637495
x=304946 possible_value=4653120
x=305970 possible_value=4668745
x=306994 possible_value=4684370
x=308018 possible_value=4699995

作者: 不是排骨不熬汤 时间: 2009-10-2 20:58:48

这个题目有点问题, 猴子先拿再除和先除猴子再拿 WILL MAKE THE DIFFERENCE.

如果猴子先拿再减1/5的话, 是3121?

(((((3121-1)X0.8-1)X0.8-1)X0.8-1)X0.8)-1)X0.8 = 1020

1020/5=204 可以最后平分.

不过这样猴子只拿了5个. 题目中猴子貌似拿了6个?

作者: 不是排骨不熬汤 时间: 2009-10-2 21:01:49

不然12495就是最小值了..

作者: love_3_month 时间: 2009-10-2 22:00:47

你的签名是今年的air show吗？

作者: 不是排骨不熬汤 时间: 2009-10-2 22:18:10

昨天阅兵空军的首批机群...

作者: celenachen 时间: 2009-10-2 22:22:26

漂亮劲的--

作者: 江南之玥 时间: 2009-10-3 03:19:11

本帖最后由江南之玥于 2013-11-16 21:27 编辑

dddddddddddddddddddddddddddddddd

作者: leiz 时间: 2009-10-3 11:21:57

本帖最后由 leiz 于 2009-10-3 11:27 编辑

是不是我没读懂题,还是这个是无解?
let n be the number of coconuts.
first man take 1/5, so n%5 == 0.
at the end, monkeys took 6 coconuts in total, so (n-6)%5 == 0.
we have (n = 0) %5 and (n = 1) %5.

编辑:没看清楚题,原来是分最后剩下的分成5份

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