A bubble chamber contains three types of subatomic particles: $10$ particles of type $X$, $11$ of type $Y$, $111$ of type $Z$.
Whenever an $X$ and $Y$ particle collide, they both become $Z$-particles. Likewise, $Y$ and $Z$ particles collide and become $X$ particles and $X$ and $Z$ particles become $Y$ particles upon collision.
Can the particles in the bubble chamber evolve so that only one type is present?
Now the book which I am referring to addresses this question using the difference of populations modulo $3$.
Let $(x,y,z)$ be the population at any time
WLOG Let there be an $X-Z$ collision. Thus the new popuation $(x-1,y+2,z-1)$
Now it can be observed that $y+2-(x-1)=y-x+3$. Since the original $y-x=1$, $D(X,Y)\equiv1 \pmod 3$ where $D(X,Y)$ is the difference in the population of $X$ and $Y$
Similarily $D(Z,Y)\equiv 1\pmod 3$ and $D(Z,X)\equiv 2\pmod 3$ which means that no two populations will ever be the same modulo $3$.
$\therefore $ At no point in time any two populations will be zero.
This was the approach from the book, but I have rather used a different approach.
My Approach:
It can be observed that: $$P(X)\equiv 1\pmod 3$$ $$P(Y)\equiv 2\pmod 3$$ $$P(Z)\equiv 0\pmod 3$$ Where $P(S)$ denotes the population of particle type $S$.
Let us observe the change in population modulo $3$ in the event of a collision.
Let there be $X-Y$ collision, thus $(1,2,0)\rightarrow (0,1,2)$ since each collision reduces the population of particles involved by $1$ and increases the population of particles uninvolved by $2$.
Similarily $Y-Z$ collision results in $(1,2,0)\rightarrow (0,1,2)$ and $Z-X$ collision results in $(1,2,0)\rightarrow (0,1,2)$
It can be observed that the change in population modulo $3$ is the same in the case of all collisions which can be explained by the fact that increasing population by $2$ means the same as reducing the population by $1$ MODULO $3$.
$\therefore $ Each collision would result in a reduction of each population by $1$ MODULO $3$ which means that the populations will always be congruent to a different number modulo $3$ no matter the type or order of collision.
Hence no pair of two populations will be the same or zero at any given time.
This was my approach. I think that I may have used the same idea as the book has used but have unnecessarily over-complicated it.
Please tell me if this approach can be used as an alternative to the one given in the book. Also please suggest any corrections in my approach if possible.
THANKS