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        riddles >> easy >> Product and Sum Triplet Product Difference Puzzle
        
(Message started by: K Sengupta on Dec 24^th, 2009, 8:16am)

Title: Product and Sum Triplet Product Difference Puzzle
Post by K Sengupta on Dec 24^th, 2009, 8:16am

Determine all possible quadruplet(s) (a, b, c, d) of positive integers with a <=b <= c <= d, that satisfy this equation:

a*b*c*d – (a*b*c + a*b*d+ b*c*d +a*c*d) = 110

Title: Re: Product and Sum Triplet Product Difference Puz
Post by R on Dec 24^th, 2009, 8:35am

But why???
:P :P

Did you forget something, like a,b,c and d are digits not integers? And abcd means a*b*c*d, if they are integers, not digits.

Title: Re: Product and Sum Triplet Product Difference Puz
Post by ThudanBlunder on Dec 24^th, 2009, 8:56am

on 12/24/09 at 08:35:37, R wrote:

Did you forget something, like a,b,c and d are digits not integers?

I think he intends them to be integers.

Title: Re: Product and Sum Triplet Product Difference Puz
Post by K Sengupta on Dec 24^th, 2009, 10:18am

on 12/24/09 at 08:35:37, R wrote:

abcd means a*b*c*d, if they are integers, not digits.

I confirm having suitably amended the problem text to obviate the ambiguity.

Title: Re: Product and Sum Triplet Product Difference Puz
Post by R on Dec 24^th, 2009, 1:26pm

If a,b,c and d are the roots of the quartic equation P₀x⁴ + P₁x³ + P₂x² + P₃x¹ + P₄ = 0, then the given condition tells us that:
P₃ + P₄ = 110 P₀
That's all I could get. There seems to be like many such quadruplets.

Title: Re: Product and Sum Triplet Product Difference Puz
Post by MathsForFun on Dec 27^th, 2009, 1:57am

I would guess that the solution lies in limits to what the divisors of a, b, c and d must be. Not sure if this is any help, but:

a = b*c*d/(((b-1)*c-b)*d-b*c)
b = a*c*d/(((a-1)*c-a)*d-a*c)
c = a*b*d/(((a-1)*b-a)*d-a*b)
d = a*b*c/(((a-1)*b-a)*c-a*b)

Title: Re: Product and Sum Triplet Product Difference Puz
Post by aicoped on Jan 25^th, 2010, 1:14pm

I have no clue how to solve this, but trial and error I got 4,5,5,6 as a solution.

EDIT

also

2,4,5,75
2,5,5,32

EDIT number 2

Ok my thoughts.

a must be bigger than 1, since you will always return a negative number with the other three answers being postive. A can't be higher than 4, since 5,5,5,5 is 125 and any other numbers will all be bigger than that. So my range is now limited to:

2<=a<=4

Once you have that, you can find maximum values for b and then test all values beneath that and repeat for c.

For example if a=2, then b can not be bigger than 7, for 2,8,8,8 yields a number higher than 110. So i place it equal to 2 find that my results will always be negative, so i set to 3 and find the same, at 4 I am finally able to get a positie answer,and then find out the maximal c value and so on.

This is very cumbersome, but I can't think of a better way.

looking at the field of answers so far:

2,4,5,75
2,5,5,32
4,5,5,6

It would seem if there are any more ansers they would have to be in the 3,x,x,x range or possibly higher than 2,5,5,x. I think I did all of them by hand but i might have missed some.