What segment in 8-bit LED Displays used for Traffic Light timers can be removed causing minimal impact in the readability of the countdown numbers?

Question

I passed by an intersection with traffic lights and noticed that 1 segment of the 8-bit display counter is dimmed (it's not working). When the lowermost segment is dimmed for example, number 4 can still be read as 4, but number 5 will not be read as 5.

First level question: I just thought if there is a mathematical way (aside from brute force method) to determine what segment could be removed from the LED display that would make minimal impact in the readability of numbers. I am curious if the solution is only 1 segment, or it can be multiple segments. And possibly the countdown timer would matter? For example, the removed segment causing minimal impact differs if the traffic light timer counts from 30 to 00 as compared to a timer that counts from 15 to 00. If we know this answer, in case 1 segment is dimmed are there are no available spare LED segments, they can just replace the dimmed LED segment from the segment with least impact in number readability.

Second level question: What if we consider in the permutations when a removed/added segment from the number will not impair readability.
For example, number 9 with and without the lowermost segment is still readable.

Thanks!

Answer 1

Let's number each segments (actually there are only 7 of them) referring to this labeling. https://upload.wikimedia.org/wikipedia/commons/thumb/0/02/7_segment_display_labeled.svg/220px-7_segment_display_labeled.svg.png

$0=a, 1=b, 2=c, 3=d, 4=e, 5=f, 6=g$

And then we can express each number as a vector $\{(v_0,v_1,v_2,v_3,v_4,v_5,v_6)\mid v_i=0\space or\space1\}$.

\begin{align} 0=(0,0,0,0,0,0,0)\\ 1=(0,1,1,0,0,0,0)\\ 2=(1,1,0,1,1,0,1)\\ 3=(1,1,1,1,0,0,1)\\ 4=(0,1,1,0,0,1,1)\\ 5=(1,0,1,1,0,1,1)\\ 6=(1,0,1,1,1,1,1)\\ 7=(1,1,1,0,0,0,0)\\ 8=(1,1,1,1,1,1,1)\\ 9=(1,1,1,1,0,1,1)\\ \end{align}

And then let's define $distance(d_{min})$ as $$d_{min}=\min d_{ij}=\min \|v_i (xor)v_j\|$$ If we don't lose any segment then the distance is $1$. The minimum distance happens for $(i,j)=(1,7),(6,8),(5,6),(8,9),(3,9)$ in which cases either segments $0$, $1$, $4$ or $5$ is the only difference. So we shouldn't lose those segments.

We can define two things.

1. $d(k)=$Minimum distance when we lost $k$'th segment.
2. $f(k)=$Impact factor of $k$'th segment $=\sum_{i,j} d_{ij}-d_{i}(k)$ where $d_{ij}(k)$ means $d_{ij}$ when $k$'th segment is lost.

Actually we can define different parameter, for example, "minimum of distance number that is changed by losing the segment" etc. But let's just keep going for now.

Then we can choose a segment that will not change $d(k)$ and has the minimum $f(k)$. I'll have to do some calculation to get this.

(*) It turns out that we could replace segments $2$, $3$, or $6$, but the segment $2$ has the minimum impact factor of $32$. The next one is the segment $6$ having impact factor of $42$. The segment $3$ has impact factor of $48$.