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A hint please.

  1. Oct 19, 2006 #1
    Can anyone give me hints on the following question.I do not know how to proceed.

    Find all integers a,b,c,d satisfying the following relations
    i) [tex]1 \leq a \leq b \leq c \leq d[/tex]
    ii) ab+cd = a+b+c+d+3

  2. jcsd
  3. Oct 20, 2006 #2


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    Staff: Mentor

    Well, since 1,2,3,4 didn't work, I'd try next to write a quick program or use Excel to see what some of the solutions look like.
  4. Oct 20, 2006 #3


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    Staff: Mentor

    Well I got at least one solution in Excel. Now if I could prove that it's the only one.....
  5. Oct 20, 2006 #4


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    Science Advisor

    I got 4 solutions using Haskell
    These are the only solutions where all values are between 1 and 20. To show that they are they only four (which I'd guess they are) you can use an argument based on how fast ab + cd grows versus how fast a + b + c + d + 3 grows.

    Incidentally, because I like to show off Haskell, this is what my code looks like
    Code (Text):

    -- f just generates all possible lists of length k where each element is at least as great as the next element.
    -- I wanted to do this efficient-like, which is why this may be a little confusing.  
    f min n 0 = [[]]
    f min n k = foldr (++) [] [[a:as | as <- (f a n (k-1))] | a <- [min..n]]

    -- If I had done it the easy way instead of the efficient way using f
    -- then then I would have just let x = [[a,b,c,d] | a<-[1..20],b<-[1..20],c<-[1..20],a*b+c*d==a+b+c+d+3 && a >= b && b >= c && c >= d]
    -- and not defined f or y
    y = f 1 20 4
    x = [[a,b,c,d] | [a,b,c,d] <- y, a*b+c*d==a+b+c+d+3]
    Then in the interpreter I just typed x.
    Last edited: Oct 20, 2006
  6. Oct 21, 2006 #5
    write the above equation like (a-1)(b-1) + (c-1)(d-1) = 5..... also, because of the first condition the second term is greater than or equal to the first term.......and since they are all positive integers..............

    can you work out the rest.....????
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