Home > Riddles > The binary bells
Four bells A, B, C, D begin silently.
There are four strings and each one changes (turns on/off) these bells:
Each rope can be pulled at most once.
In the end, only A and D are playing.
What ropes were pulled?
Back to problem
Answer: X, Y and W (and not Z) were rolled.
We model each string with binary variable ($1$ = pulled, $0$ = not).
Final objective: $A=1,\ B=0,\ C=0,\ D=1$.
Module 2 equations:
$$ \begin{aligned} A&=X+Z=1,\\ B&=X+Y=0,\\ C&=Y+Z+W=0,\\ D&=Z+W=1. \end{aligned} $$
From $D$: $W=1+Z$.
In $C$: $Y+Z+(1+Z)=0\Rightarrow Y+1=0\Rightarrow Y=1$.
In $B$: $X+1=0\Rightarrow X=1$.
In $A$: $1+Z=1\Rightarrow Z=0$.
So $W=1$.
Only solution:
$$ (X,Y,Z,W)=(1,1,0,1). $$
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