Nim Game: Find Proof for D > A, E > B, F > C

[unscientific]

Hi all, ok there are 3 piles containing objects; Pile A, Pile B and Pile C.

Let the Nim-sum of all 3 piles be denoted as X.

Let Pile A and Pile B and Pile C contain objects.
( In Nim-Sum )

A + X = D

B + X = E

C + X = F

As we are currently in research of a math project, we are wondering if there is a case whereby D > A, E > B, and F > C. If not, what is the proof?

 

[HallsofIvy]

Perhaps I am not understanding the question. The number of objects in each pile are A, B, C respectively. What do you mean by "Nim-sum"? Certainly if X> 0 then it must be the case that D> A, E> B, and F> C.

 

[CRGreathouse]

Hi all, ok there are 3 piles containing objects; Pile A, Pile B and Pile C.

Let the Nim-sum of all 3 piles be denoted as X.

Let Pile A and Pile B and Pile C contain objects.
( In Nim-Sum )

A + X = D

B + X = E

C + X = F

As we are currently in research of a math project, we are wondering if there is a case whereby D > A, E > B, and F > C. If not, what is the proof?

I don't know what a Nim-Sum is, but from what you have
$$D>A\Leftrightarrow E>B\Leftrightarrow F>C\Leftrightarrow X>0$$

so you're just asking if the Nim-Sum is always negative. Well, define it for us and we'll see!

 

[Werg22]

According to wikipedia, a nim-sum is an operation used in game theory.