# Thread: Can this be done?

1. ## Can this be done?

a + b + c + d = a * b * c * d = 1

What are a, b, c and d?

2. ## Re: Can this be done?

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>It's not so hard using complex numbers (of the form m+n*i, where i is the square root of -1), but you haven't specified what kind of numbers are allowed.</font color=yellow></span hi>

3. ## Re: Can this be done?

Please use any types of numbers you would like.

4. ## Re: Can this be done?

OK, then:

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>a = 1, b = 1, c = (-1 - i*SQRT(3))/2, d = (-1 + i*SQRT(3))/2 where SQRT(3) is the square root of 3 and i is the imaginary number representing the square root of -1.</font color=yellow></span hi>

5. ## Re: Can this be done?

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>Or a solution with real numbers (i.e. not involving the imaginary number i):

a = b = (1 + SQRT(17))/4, c = d = (1 - SQRT(17))/4</font color=yellow></span hi>

6. ## Re: Can this be done?

Hans, <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>Wouldn't it be reasonable to assume that "a" and "b" as different varaibles could not be equal to each other? Surely, within the confines of a puzzle, "a" and "b" would represent two different numbers. Otherwise, the puzzle writer would've presented a + a + c + c = a * a * c * c = 1</font color=yellow></span hi>

But maybe you're correct. It's not like I have a better solution. <img src=/S/smile.gif border=0 alt=smile width=15 height=15>

7. ## Re: Can this be done?

Ricky,

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>In mathematics, letters such as a, b, x or y are used to indicate variables, they can be equal or not equal. Within the limits of a puzzle, different letters often stand for different values, but I think this should be stated explicitly. I don't know anything about the background of this puzzle, but I doubt that the person who designed it had complex/imaginary numbers in mind, perhaps not even algebraic numbers, I'm still looking for a solution involving rational numbers (fractions) only. Timbo allowed us to use any kind of numbers, however.</font color=yellow></span hi>

8. ## Re: Can this be done?

If you want them all to be different:
<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>a = 2, b = -1, c = sqrt(2)/2, d = -sqrt(2)/2</font color=yellow></span hi>

This type of pattern can be used to get an inifinite number of answers all with them with different a,b,c,d' s
<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>Pick a (non-zero) number for a. b = 1-a. c = 1/sqrt(a*[img]/forums/images/smilies/cool.gif[/img] , d = -1/sqrt(a*[img]/forums/images/smilies/cool.gif[/img]

c+d = 0 , a+ b = 1 so a+b+c+d = 1
c*d = 1/(a*[img]/forums/images/smilies/cool.gif[/img] so a*b/(a*[img]/forums/images/smilies/cool.gif[/img] = 1</font color=yellow></span hi>

Steve

9. ## Re: Can this be done?

Here are some solutions using fractions:

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>a = 3/2, b = 4/3, c = -1/3, d = -3/2
a = 5/6, b = 9/5, c = -4/5, d = -5/6
a= 8/3, b = 9/8, c = 1/8, d = -8/3</font color=yellow></span hi>

(plus permutations, of course)

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