# Aid solving logical puzzle

I have a logical puzzle here that I need to solve. I'm not after just the answer of this puzzle, but also the reasoning and thought-process behind solving it. Any assistance is greatly appreciated.

What number between 1 and 7 do the following equal? A=, B=, C=, D=, E=, F=, G=

Given that:

1.  A ≠ 2
2.  A + B = F
3.  C – D = G
4.  D + E = 2F
5.  E + G = F


The rules are:

1. All the variables (A, B, C, D, E, F, G) are equal to integer values between 1 and 7
2. None of the variables (A, B, C, D, E, F, G) are equal to each other i.e. all seven values will be used, no repeat use of integers

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Notice that if you have positive numbers, the relation $x+y=z$ implies $x<z$ and $y<z$. So properties 2., 3., 5. give $A<F$, $B<F$, $D<C$, $G<C$, $E<F$, $G<F$. Moreover since $D+E = 2F$ and $E<F$ you get $F<D$. So $A,B,E,G < F < D < C$ which means that $F = 5, D = 6, C = 7$.

Then from 4. $E=4$, from 3. $G=1$ from 1. $A=3$, $B=2$.

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I'm confused about "Then from 4. E=4, from 3. G=1 from 1. A=3, B=2. " How did you discover E > G etc ...? –  Ospho Dec 4 '13 at 8:18
4. says $D+E=2F$ and since you know $D$ and $F$ you can solve for $E$. The same in 3. –  Emanuele Paolini Dec 4 '13 at 8:24

$A = 3, B = 2, C = 7, D=6, E = 4, F = 5, G=1$

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I didnt want the answer only, How did you come to this conclusion? Please explain –  Ospho Dec 4 '13 at 7:42
I wish I could tell you... it's the first thing I tried, and I guarantee there's a more targeted approach than what I did. Clearly $F$ is the most critical number. So I chose $F = 5$, a number somewhere in the middle, and went from there. –  Euler....IS_ALIVE Dec 4 '13 at 7:46
@Euler....IS_ALIVE: Might be you can expand your answer a little bit so that it is more useful to the OP and others who are interested. –  John Ma Dec 4 '13 at 7:56