Could someone please explain the voting system for this series.

Obviously the judges scores give a listing and ranking but how does the public vote affect the final results. What if the public voted the least for a couple that got top judges marks?

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The judges leader board makes up half of the points allotted to them, the public leader board makes up the other half. So say they were placed thus on the judge's leader board:

1st Kara - 5 points
2nd Matt - 4 points
3rd Scott - 3 points
4th Pamela - 2 points
5th Gavin - 1 point

And when the votes were tallied, they were placed here on the public vote leader board ...

1st (most votes) Gavin - 5 points
2nd Matt - 4 points
3rd Kara - 3 points
4th Scott - 2 points
5th Pamela - 1 point

... when they're all added together it would be

Kara: 5+3 = 8
Matt: 4+4 = 8
Scott: 3+2 = 5
Pamela: 2+1 = 3
Gavin: 1+5 = 6

So it would be Pamela and Scott in the bottom two with Pamela leaving. If the two in the bottom two are level on combined points, whoever got more votes than the other goes through and the other leaves. It's entirely possible for someone to come first on the judges leader board and still go out and it's also entirely possible for someone to come first on the public leader board and still go out. When there's less dancers in the competition (and especially when there's ties) it all becomes very messy and the chance of someone leaving, no matter how good they are, is always there.

It's very straightforward actually but that doesn't stop it from pickling my head a little lol.

Indeed, it's an interesting conclusion.

If you have a minimum 1 from the judge vote and you have 5 from the public that's 6 total. It's entirely possible for three others to get 7 points and you're out.

A 5+2=7
B 4+3=7
C 3+4=7
D 2+1=3
E 1+5=6

E could get millions of votes, a handful of votes could determine the order of A, B and C but that handful sends E out If D gets 2 or 3 public votes they still go out but E doesn't The conclusion is that everyone voting for E must also vote for D whatever their performance standard to ensure they get more than 1 public vote.

I'm sure that's not what was intended.

It's somewhat similar to the Tom, Rachel, Lisa semi final. Lisa and Rachel tied on 3 judge points, Tom had 1. In that series, Tom must be in the dance off so his exit was entirely the judges decision when the public were still being encouraged to vote for him. The vote was cancelled, all three went to the final and the producers were revealed as con artists or more realistically innumerate bumpkins.

Not quite the same here in that there is no dance off. However it should be pointed out to any potential voter that masses of votes for whoever finishes last with the judges will not save them when some voting results for the others can still send them out. The only way to ensure your candidate survives is to vote for someone else too. The BBC has never (as far as I know) confirmed that multiple votes all count. Neither have they ever encouraged multiple votes as far as I recall, but it is required this year.

I wonder if anyone at the regulator needs to have a word with the producers?

The same is, of course, true in the quarter finals.

It's set up so that, in the later stages of the series, the judges can over ride the public vote and keep their own favourites in.

Not quite the same. In the quarter final or any previous round only the bottom placed goes out. If you're top with the public you score at least average points. At least one other must score fewer than average or the average with fewer public points. Any contestant with the highest public vote will never finish bottom overall but they might finish second from bottom however much public support they have. Only the semi final differs by losing two which can condemn the bottom placed contestant with the judges unless you consider voting for other contestants than the one you want to keep in.

This year BBC is not in danger of the 3-3-1 fiasco two years ago. Then Tess encouraged viewers "to vote and save your couple from the dance-off". Tom was on 1 and it was mathematically impossible to vote and save him from the dance-off, so Tess was misleading. The BBC called off the dance-off and refunded phone costs to all who applied.

This year at all stages your vote COULD theoretically save your couple from elimination. That possibility is sufficient to get BBC off the hook with regulators.