Wikipedia:Reference desk/Archives/Mathematics/2009 September 18

= September 18 =

Substitution in the integral
Suppose we have the integral of the form $$\int_{0}^{h(x)}f(g(t))dt$$ and then we do a substitution of the form $$g(t)=u$$. what will happen for the integral limits, 0 and $$h(x)$$ ? —Preceding unsigned comment added by Re444 (talk • contribs) 15:32, 18 September 2009 (UTC)
 * I think of it by imagining a little "t=" next to each limit. You want those to turn into "u=", so figure out what u is when t=0, and similarly when t=h(x). Does that do it? -GTBacchus(talk) 15:36, 18 September 2009 (UTC)
 * Oh thanks, I really need that! It solved all my problems. Re444 (talk) 18:53, 18 September 2009 (UTC)

A really brainy sum !!!!!
My friend sent me this question via mail and i tried various options for getting this sum.Even after 3 hrs of futile trying I failed pls help.The question as follows

A farmer was dividing his property among his children. This was what the farmer told to his first son “Take as many number of cattle as you can care for and your wife may take one-ninth of the remaining number.” To his second son he said “Take one more than what the first son took and your wife will have one ninth of the cattle remaining after you have taken.” This applied to the remaining sons too i.e. each son would take one more than the next oldest brother and their wives would have one ninth of the remaining after their husbands have got their share. After the cattle were divided, the farmer proceeded to divide the gold bars he had among the children. Each gold bar was valued at 3.5 times each cow's value. The gold bars were divided such that each couple had equally valued inheritance. What are the number of cows, gold bars and the number of sons the farmer had?

My deductions : so pls help...3 to 4 hrs of working on this didnt help much :-(164.100.170.4 (talk) 16:14, 18 September 2009 (UTC)
 * 1) Obviously the wife of last son will not get any cows as there should be no cow left.
 * 2) The number of cows of each couple consists of three terms.All terms except the constant term follow a particular pattern except the integer.
 * 3) The difference between the number of cows(assuming the cost of a cow is 1 unit ) of each couple should be equal to a multiple of 7.
 * Does the wife round down or up?83.100.251.196 (talk) 16:56, 18 September 2009 (UTC)
 * The wives don't round. That's part of the problem, is that there's always a multiple of nine left when the wife chooses. Nice problem. This'll take a minute... -GTBacchus(talk) 19:37, 18 September 2009 (UTC)

what does that mean???


 * Simplest non-trivial solution: 2 sons, 135 cows, 2 bars of gold. The first son takes 63 cows, leaving 72. His wife takes 8. The second son gets the remaining 64 cows, and the second wife, none. That's 71 cows for the first couple, and only 64 for the second. The 2 bars of gold make it even. Of course, if there's only 1 son, it's trivial, and (2 sons, 16 cows, no gold) works, but isn't very interesting. There are other solutions, with 2 sons. Not sure about 3 or more. -GTBacchus(talk) 20:13, 18 September 2009 (UTC)

(after ec) The problem has several possible solutions. For example: I am confident that there are other solutions with more sons, but I haven't worked them out yet. Note though, that the number of gold bars will always be non-unique, because for any solution with n sons, one can always add np to the number of gold bars while meeting all the conditions of the problem. Abecedare (talk) 20:15, 18 September 2009 (UTC)
 * Number of sons = 1; Number of cows = arbitrary; number of gold bars = arbitrary. The first son takes all the cows, and his wife gets none.
 * Number of sons = 2; Number of cows = 16; Number of goldbars = any even number, say 2p. First son takes 7 cows, his wife gets (16-7)/9 = 1; second son gets 7+1=8, and his wife gets none. Both couples have 8 cows each, so the gold bars are divided equally.
 * Ah, good point about the gold. You can always throw more gold around, as long as the extra is divisible by the number of couples. For more unique solutions, one would add the condition that the couple with the most cows gets no gold. -GTBacchus(talk) 20:18, 18 September 2009 (UTC)

There are no solutions with 3 sons. The number of cows the first son takes would have to be congruent to both 6 and 0, modulo 64, which is impossible. -GTBacchus(talk) 20:24, 18 September 2009 (UTC)
 * I can post the general solution for 2 sons, but I don't want to ruin any more of the fun for others... feel free to post to my talk page for it. -GTBacchus(talk) 20:28, 18 September 2009 (UTC)

Hmmm, I found a few solutions with three sons! I am guessing there are solutions with even more number of progeny, but I'm donw with this family :-) Abecedare (talk) 20:40, 18 September 2009 (UTC)
 * Number of cows = 24; Goldbars = 3p; First son takes 6 cows, and his wife takes (24-6)/9=2; Second son takes 7 cows and his wife takes (24-6-7-2)/9=1; Third son takes the remaining 8 cows. Each couple has 8 cows, and can divide the gold evenly.
 * Number of cows = 1543; Gold bars = 3p+52. First son takes 454 cows, and his wife gets (1543-454)/9 =121 cows. Second son takes 455 cows and his wife gets (1543-454-455-121)/9 =57 cows. Third son takes the remaining 456 cows. So first couple has 575 cows, second couple 512 cows and third couple has 456 cows. So first couple gets p goldbars, second couple p+18 goldbars, and third couple p+34 goldbars, to even the inheritence.
 * Huh, what did I do wrong? Runs off to double-check congruences... -GTBacchus(talk) 20:42, 18 September 2009 (UTC)
 * Aaargh. How silly of me. Thanks for the down-a-peg; those are good for the soul. -GTBacchus(talk) 20:44, 18 September 2009 (UTC)


 * (ec) It's easier to run it in reverse. Start with no cows on the pile.  First wife adds no cows, first son (last really) adds some number of cows that has to be a multiple of 8, call it 8k, so that the second wife can add 1/8 of the existing pile.  The second son adds 8k-1, and the new pile (with 17k-1) now has to be a multiple of eight for the third wife, etc.  The only exception is that the last son (first really) doesn't need to make the pile a multiple of 8.  For each additional son that there is in the family, there are stricter constraints, but always an infinite number of solutions.  For example with 2 son there are no constraints on k.  With 3 sons, k = 1(mod 8).  With 4 sons k = 1(mod 64).  With n sons, k = 1(mod 8n-2) I think (it's definitely a sufficient condition, but it might be looser than that).  Then there's the gold bar thing which adds some more constraints.  1 son puts no constraints on k.  With 2 sons, k = 1(mod 7).  I'm less sure about how this one grows with the number of sons, although I think requiring k = 1(mod 7n-1) is always safe.  So combining these, for n sons, requiring k = 1(mod 7*56n-2) should always get you solutions I think, but I'm not totally sure.
 * Although there are infinite solutions for any given number of sons, an easy solution that pops out is k=1. Each couple gets 8 cows, and no gold is required.  That would work for any number of sons, except that the cows each son takes has to be non-negative, so it fails for more than 9 sons. Rckrone (talk) 21:19, 18 September 2009 (UTC)
 * That is easier. I made my variable the number of cows that the first son takes, which took a bit more calculating. -GTBacchus(talk) 21:26, 18 September 2009 (UTC)
 * With no rounding I came to the conclusion that there could be any number of sons - though clearly as 'sons' increase the number of cattle increases vaguely 'factorially' (to satisfy the modular relationships). The gold bars don't seem to affect it at all (at any number of sons I can multiply all cattle by 7 to get a integral relationship with number of gold bars..) Did anyone find a reason not to suggest any number of sons?
 * the issue I have is that there doesn't seem to be a proper answer 83.100.251.196 (talk) 22:00, 18 September 2009 (UTC)


 * "at any number of sons I can multiply all cattle by 7" : That's not true since the condition that each son recieves one more cow than his elder brother would then not be satisfied.
 * The analysis by Rcrone above is right on the point: we need to simultaneously satisfy congruency relations with respect to some power of 8 and some of power of 7, and since this is achievable for any power n, (infinitely many) solutions exist for any given number of sons. Abecedare (talk) 22:10, 18 September 2009 (UTC)
 * Yes, I just came back to correct that.83.100.251.196 (talk) 22:34, 18 September 2009 (UTC)
 * I think with last wife taking no cows the method fails after 2 sons, (though probably made a mistake) I did. Doing sons wrong way round.83.100.251.196 (talk) 22:50, 18 September 2009 (UTC)
 * cows=16 sons=2
 * first son = 7, wife = 1
 * second son 8, wife =0
 * cow difference = 0, thus no gold
 * No solution for 3 sons I can see. (reason:algebraic (5k-1)/8 must equal integer for integer k - impossible)
 * Solutions also exist for cows = 16+34n, fs=16n+7,fw=2n+1,ss=16n+8,sw=0, cow difference=2n = 7n gold.
 * Ignore 1 son solution since it seems irrelevent to the question. Is that ok?83.100.251.196 (talk) 23:30, 18 September 2009 (UTC)
 * Doh, me dumbo see correct solution below.83.100.251.196 (talk) 00:09, 19 September 2009 (UTC)
 * (5k-1)/8 is an integer if k=5+8n for any integer n. In that last solution, remember, ss=fs+1.


 * General solution for 2 sons: let k be any non-negative integer.
 * Cows: 16+119k.
 * Gold: 2k.
 * Son 1 takes: 7+56k cows.


 * General solution for 3 sons: let k be any non-negative integer.
 * Cows: 24+1519k.
 * Gold: 52k.
 * Son 1 takes: 6+448k cows.


 * General solution for 4 sons: let k be any non-negative integer.
 * Cows: 32+17225k.
 * Gold: 902k.
 * Son 1 takes: 5+3584k cows.
 * -GTBacchus(talk) 23:51, 18 September 2009 (UTC)
 * Can anyone write it down for n sons? I'm working on 5... -GTBacchus(talk) 23:56, 18 September 2009 (UTC)


 * Empirically observed pattern: first son's take when there are n sons = (9-n) + (8^(n-1))*7k. One can trace back the total number of cows from this, but I'm too lazy to do that at the moment. My conjecture is that this pattern will hold for n<9, and possibly for n=9. Abecedare (talk) 03:36, 19 September 2009 (UTC)