[Guest]

Hello:

Could someone tell me if it is possible to find the value of ten variables (A,B,C,D,E,F,G,H,I,J) that fulfill the following restrictions:

A > 0,

B > 0,

C > 0,

D > 0,

E > 0,

F > 0,

G > 0,

H > 0,

I > 0,

J > 0,

A = 1,

J = 100,

A+C > B,

A+D > B+C,

A+E > B+C+D,

A+F > B+C+D+E,

A+G > B+C+D+E+F,

A+H > B+C+D+E+F+G,

A+I > B+C+D+E+F+G+H,

A+J > B+C+D+E+F+G+H+I,

B+D > A+C,

B+E > A+C+D,

B+F > A+C+D+E,

B+G > A+C+D+E+F,

B+H > A+C+D+E+F+G,

B+I > A+C+D+E+F+G+H,

B+J > A+C+D+E+F+G+H+I,

C+D > A+B,

C+E > A+B+D,

C+F > A+B+D+E,

C+G > A+B+D+E+F,

C+H > A+B+D+E+F+G,

C+I > A+B+D+E+F+G+H,

C+J > A+B+D+E+F+G+H+I,

D+E > A+B+C,

D+F > A+B+C+E,

D+G > A+B+C+E+F,

D+H > A+B+C+E+F+G,

D+I > A+B+C+E+F+G+H,

D+J > A+B+C+E+F+G+H+I,

E+F > A+B+C+D,

E+G > A+B+C+D+F,

E+H > A+B+C+D+F+G,

E+I > A+B+C+D+F+G+H,

E+J > A+B+C+D+F+G+H+I,

F+G > A+B+C+D+E,

F+H > A+B+C+D+E+G,

F+I > A+B+C+D+E+G+H,

F+J > A+B+C+D+E+G+H+I,

G+H > A+B+C+D+E+F,

G+I > A+B+C+D+E+F+H,

G+J > A+B+C+D+E+F+H+I,

H+I > A+B+C+D+E+F+G+H,

H+J > A+B+C+D+E+F+G+H+I,

I+J > A+B+C+D+E+F+G+H,

Thanks in advance.

Edited September 30, 2011 by mselva

[CaptainEd]

These two lines seem a little out of pattern

H+I > A+B+C+D+E+F+G+H,

H+J > A+B+C+D+E+F+G+H+I,

Given the pattern above, I would expect

H+I > A+B+C+D+E+F+G

H+J > A+B+C+D+E+F+G+I

However, it's your puzzle, so I'm simply asking, is this discrepancy intentional, or is it a typographical error?

[CaptainEd]

also, are the values required to be integer?

[CaptainEd]

I'm pretty sure it's not possible. if A=B=C=1, each of the rest need to be double the previous, making J=128.

represent a tiny fraction with "e"

A=B=C=1

D=1+e

E=2+2e

F=4+4e

G=8+8e

H=16+16e

I=32+32e

J=64+64e

[Guest]

here is the answer

B=0.1

C=0.2

D=0.3

E=0.6

F=1.2

G=2.4

H=4.8

I=9.6

[Guest]

here is the answer

B=0.1

C=0.2

D=0.3

E=0.6

F=1.2

G=2.4

H=4.8

I=9.6

But then B+H=4.9 while A+C+D+E+F+G = 5.7 violating the line "B+H > A+C+D+E+F+G," Other conflicts also.

We can be assured that there is no integer solution If decimals are allowed, I believe that a solution exists - will work on it more tomorrow.

[Guest]

These two lines seem a little out of pattern

H+I > A+B+C+D+E+F+G+H,

H+J > A+B+C+D+E+F+G+H+I,

Given the pattern above, I would expect

H+I > A+B+C+D+E+F+G

H+J > A+B+C+D+E+F+G+I

However, it's your puzzle, so I'm simply asking, is this discrepancy intentional, or is it a typographical error?

Yes Captain Ed, you are right, I have a error in the lines h+i and h+j. You proposal is right.

If there is not integer solution, which would be the minimum value of "J" to use just integer numbers?

Thanks

[Guest]

These are the minimum values for the variables when restricted to integers;

A 1

B 1

C 1

D 2

E 4

F 8

G 16

H 32

I 64

J 128

By further restricting no two variables have same value:

A 1

B 2

C 3

D 5

E 10

F 29

G 49

H 98

I 196

J 392

[Guest]

If the values of A -> J can be the same as each other, and also if you are allowed real numbers or fractions, the following works perfectly:-

A 1

B 1

C 1

D 1.5

E 3

F 6

G 12

H 25

I 50

J 100

[Guest]

I was trying to find out how far off a solution with only integers we were and the following combination is so close...

If the greater than was greater than or equal it would work

A 1

B 1

C 1

D 2

E 3

F 6

G 12

H 24

I 48

J 100

[Guest]

If we allow decimals, then there are many solutions- we could even reduce the value of J. Given A=1 and allowing 1 decimal place, the smallest values possible

A 1

B 1

C 1

D 1.1

E 2.2

F 4.4

G 8.8

H 17.6

I 35.2

J 70.4

Also, testing various values I found

No incident where a value satisfying the first 8 inequalities failed to satisfy the remaining inequalities.

If we require integers but allow greater than or equal to, then

A 1

B 1

C 1

D 1

E 2

F 4

G 8

H 16

I 32

J 64

I am showing minimum value for J though any greater value still satisfies equations.

Edited October 3, 2011 by thoughtfulfellow