frederic@ubvax.UUCP (Frederic Bach) (05/24/85)
I once won a bottle of Champagne by solving this rather amusing problem. There once were two brothers herding a flock of sheep. One day, they decided to sell them all in the market. Each beast was sold for as many dollars as sheep there originally were in the herd. The total amount of money was in $10 bills, plus less than $10 in $1 bills. The elder brother proceded to share it. He took a $10 bill, then gave one to his brother, then took one again, and so on till he turned out taking the last $10 bill ; he then gave all of the $1 bills to the youngster, who burst out : "Hey, but you've got #@% dollars more than I have ". How about now working out what he really said ?
dkatz@zaphod.UUCP (Dave Katz) (05/28/85)
In article <179@ubvax.UUCP> frederic@ubvax.UUCP (Frederic Bach) writes: > There once were two brothers herding a flock of >sheep. One day, they decided to sell them all in >the market. Each beast was sold for as many dollars >as sheep there originally were in the herd. >The total amount of money was in $10 bills, plus >less than $10 in $1 bills. The elder brother proceded >to share it. He took a $10 bill, then gave one to >his brother, then took one again, and so on till >he turned out taking the last $10 bill ; he then >gave all of the $1 bills to the youngster, who burst >out : "Hey, but you've got #@% dollars more than I have ". > >How about now working out what he really said ? FOUR As described, there must have been an odd number of $10's. Since the total value is the square of the number of sheep, look for values of 2 <= y <= 11 for which int((y^2) / 10) is odd, (ignoring the trivial case of 1 since the problem implies there were at least $30). The only cases are 4^4=16, 6^6=36. All others are even. Further, (((n*10)+y)^2) = (n*10)^2 + 2ny + (y^2). Since int(((n*10)^2 + 2ny) /10 ) is always even, ((n*10)+y)^2 is odd in the second digit iff y^2 is odd in the second digit. This applies for all n | 0 <= n (there being a positive number of sheep). Hence the number of sheep must have been: (n*10+y | 0 <= n, y = 4 or y = 6) In either case, the value ends in 6 and the difference is four - Now, where is my bottle of champagne?
js2j@mhuxt.UUCP (sonntag) (05/28/85)
> There once were two brothers herding a flock of > sheep. One day, they decided to sell them all in > the market. Each beast was sold for as many dollars > as sheep there originally were in the herd. The total amount of money was a square number, then. > The total amount of money was in $10 bills, plus > less than $10 in $1 bills. The elder brother proceded > to share it. He took a $10 bill, then gave one to > his brother, then took one again, and so on till > he turned out taking the last $10 bill ; There were an odd # of $10 bills plus change. > he then > gave all of the $1 bills to the youngster, who burst > out : "Hey, but you've got #@% dollars more than I have ". > > How about now working out what he really said ? The amount of money they had was a square number with an odd digit in the 10's place. Searching the numbers less than 10 exhaustively, we find that 4 and 6 are the only numbers which, when squared, yield such numbers. The numbers greater than 10 can all be written as N=10*a+b, where a and b are integers less than 10. When squared, N^2=100*a^2+20*a*b+b^2. Notice that the first two terms of this expression have an *even* number of tens. The only way N^2 can have an odd digit in the 10's place is if b^2 has an odd digit there. Thus the number of sheep (and the price of the sheep) must be in the following sequence: 4,6,14,16,24,26,34,36... Happily, the square of any number in this series ends in the same digit, 6, which is the number of $1 bills which the younger brother receives in lieu of a tenner. So what he really said was "Hey, but you've got *four* dollars more than I have!" Now where's my champagne? *** REPLACE THIS LINE WITH YOUR MESSAGE *** -- Jeff Sonntag ihnp4!mhuxt!js2j "You can be in my dream if I can be in yours." - Dylan
weissler@randvax.UUCP (Robert Weissler) (05/31/85)
I thought I'd try my hand at the following puzzle. > There once were two brothers herding a flock of > sheep. One day, they decided to sell them all in > the market. Each beast was sold for as many dollars > as sheep there originally were in the herd. > The total amount of money was in $10 bills, plus > less than $10 in $1 bills. The elder brother proceded > to share it. He took a $10 bill, then gave one to > his brother, then took one again, and so on till > he turned out taking the last $10 bill ; he then > gave all of the $1 bills to the youngster, who burst > out : "Hey, but you've got #@% dollars more than I have ". > > How about now working out what he really said ? He really said: "Hey, but you've got 6 dollars more than I have." Here's my approach to the solution: There are n sheep, where each sheep sells for n dollars. According to the 3rd sentence above, that means the total amount of money the brothers made was n**2 dollars. This money was in m $10 bills and less than 10 (say, k) $1 bills, so: 10m + k = n**2 or n**2 - 10m - k = 0. Since the elder brother took the first and last $10 bill, m must be odd. Being the math klutz that I am, I had to solve it for k by trying several values for n and m. Since we are talking dollars, n, m and k must be integers. So: Let m = 1, 3, 5, 7...(the odd integers) Then 10m = 10, 30, 50, 70... So n**2 must be greater than 10m by less than 10 (i.e., k). Thus n**2 = 16, 36, -, 64 (25 and 49 don't work) And finally k = n**2 - 10m = 6 in each case above. I'm sure the more mathematically adept can find a really elegant solution, but I'll let somebody else waste their time on that. (:-) -Robert ARPA: Weissler@Rand-unix UUCP: randvax!weissler
thomas@utah-gr.UUCP (Spencer W. Thomas) (06/01/85)
<enter mild flame mode>
I sure am glad I took a minute to work this problem before reading the
next article (the one this message is a followup to). Otherwise I
wouldn't have had the pleasure of discovering the answer for myself.
*PLEASE, PLEASE, PLEASE*, if you must show off your puzzle-solving ability
by posting the answer to the net, at least encrypt the solution (and add
a "(SPOILER)" to the header, for extra warning.) Judging from the
number of wrong answers I have seen posted to this newsgroup (and
others), you may be better off NOT posting the answer at all.
Don't let us know what a jerk you are that you have to be the first one
on the net with "the answer". Show some consideration.
<somewhat relieved>
--
=Spencer ({ihnp4,decvax}!utah-cs!thomas, thomas@utah-cs.ARPA)
"A pupil from whom nothing is ever demanded which he cannot do
never does all he can." -- John Stuart Millsml@luke.UUCP (Steven List) (06/01/85)
> In article <179@ubvax.UUCP> frederic@ubvax.UUCP (Frederic Bach) writes: > > > There once were two brothers herding a flock of > >sheep. One day, they decided to sell them all in > >the market. Each beast was sold for as many dollars > >as sheep there originally were in the herd. > >The total amount of money was in $10 bills, plus > >less than $10 in $1 bills. The elder brother proceded > >to share it. He took a $10 bill, then gave one to > >his brother, then took one again, and so on till > >he turned out taking the last $10 bill ; he then > >gave all of the $1 bills to the youngster, who burst > >out : "Hey, but you've got #@% dollars more than I have ". > > > >How about now working out what he really said ? > > FOUR > > As described, there must have been an odd number of $10's. Since the > total value is the square of the number of sheep, look for values of > 2 <= y <= 11 for which int((y^2) / 10) is odd, > (ignoring the trivial case of 1 since the problem > implies there were at least $30). The only cases are 4^4=16, 6^6=36. > All others are even. > > Further, (((n*10)+y)^2) = (n*10)^2 + 2ny + (y^2). Since > int(((n*10)^2 + 2ny) /10 ) is always even, ((n*10)+y)^2 is odd in the > second digit iff y^2 is odd in the second digit. This applies for all > n | 0 <= n (there being a positive number of sheep). > > Hence the number of sheep must have been: > (n*10+y | 0 <= n, y = 4 or y = 6) > In either case, the value ends in 6 and the difference is four - > > Now, where is my bottle of champagne? In reading the above explanation and solution, I agree with the answer. FOUR. However, I didn't bother with all the advanced algebra. If you step through the first few squares, you will find that 4, 6, and 16 all produce squares that (1) are odd multiples of 10 and (2) end in 6. Since the total sale price must be a square (each sheep sold for a price equal to the total number of sheep), and at least three answers end in 6 (producing a remainder of four), why go on and on and on... Can I have a glass? (:->
js2j@mhuxt.UUCP (sonntag) (06/03/85)
> I sure am glad I took a minute to work this problem before reading the > next article (the one this message is a followup to). Otherwise I > wouldn't have had the pleasure of discovering the answer for myself. > *PLEASE, PLEASE, PLEASE*, if you must show off your puzzle-solving ability > by posting the answer to the net, at least encrypt the solution (and add > a "(SPOILER)" to the header, for extra warning.) Could I get some feedback from others on this subject? I always assume that if the title of an article says 'Re: How to confuse Penguins' that the article very well might be the *answer* to the 'How to confuse Penguins' puzzle. I've seen very few people posting spoiler warnings or rotating answers to puzzles, and personally have never missed them. But then again, I usually *do* take a minute to work out the interesting puzzles I find here immediately, and usually don't need such warnings anyhow. Are there more people out there who think solutions to puzzles should have warnings and be rotated? Howabout just a warning? -- Jeff Sonntag ihnp4!mhuxt!js2j "Sundown, yellow moon. I replay the past. I know every scene by heart; they all went by so fast." - Dylan
edhall@randvax.UUCP (Ed Hall) (06/05/85)
> <enter mild flame mode> > > I sure am glad I took a minute to work this problem before reading the > next article (the one this message is a followup to). Otherwise I > wouldn't have had the pleasure of discovering the answer for myself. > *PLEASE, PLEASE, PLEASE*, if you must show off your puzzle-solving ability > by posting the answer to the net, at least encrypt the solution (and add > a "(SPOILER)" to the header, for extra warning.) > -- > =Spencer ({ihnp4,decvax}!utah-cs!thomas, thomas@utah-cs.ARPA) A simple guide-line: if you want to solve a puzzle yourself, skip over all the ``Re:'' postings to that subject. In this group the ``Re:'' is often pronounced ``Spoiler:''. Maybe the next version of netnews should offer to ROT13 replies to net.puzzle the same way it offers to ROT13 net.joke submissions. -Ed Hall decvax!randvax!edhall