The maths that proves nobody is safe from the Strictly dance-off

Not even the location of a lost remote control can create more sofa tension on a Sunday evening than the latest Strictly Come Dancing result.

The culmination of a week’s hard work by the competing couples will see one of them leave the popular BBC One contest, through a combination of the judges’ scores, the public vote and, ultimately, the performance in the dreaded dance-off.

After a Saturday evening of eye-catching and technically demanding foxtrots, waltzes, Charlestons, American smooths and couple’s choices, there is usually one pair so accomplished they couldn’t possibly be in danger from elimination.

Except they are. And there’s maths to prove it.

Call in the experts

Norman Biggs is emeritus professor of mathematics at the London School of Economics. He has written previously about the intriguing combinations of judge and viewer votes, which have placed some of the best dancers in peril and saved those who didn’t impress the panel so much.

Prof Biggs shared his findings with BBC Bitesize, beginning with the Saturday night scores from the four Strictly judges. He explained: “The judges award the contestants scores out of 40. The scores are then converted into a ranking list of points. If there are six contestants and no ties, the list would be: 6, 5, 4, 3, 2, 1.”

It’s where the scores tie that some interesting quirks are thrown up. The Strictly rules state that tied scores receive the same ranking points. So, in this six-couple example, say two pairs tied in second place - they would each receive five points. The couple immediately below them receives the ranking score of four – normally reserved for third place despite those dancers not actually finishing third.

Prof Biggs continued: “If there are ties, the rules might produce, for example: 6, 5, 5, 4, 3, 2.”

This shows a tie for second place, but as Prof Biggs noted, it may be more logical to award ranking scores (in this instance) of 6, 5, 5, 3, 2 and 1. This would see the fourth couple in the list receive the three points equivalent with fourth place and the couple with the lowest score earn a solitary point for being last.

And then the public score is added.

The public has spoken

Prof Biggs said: “The public vote produces a similar list, but because the number of votes is large, it is practically impossible for ties to occur, so the list always contains all the numbers from 6 to 1 in some order.”

Here’s one example of a combined judges and public vote, once the viewer scores have been converted into the same ranking points as the judges’ were on Saturday night.

COUPLEJUDGESVIEWERSTOTALRANKINGDANCE-OFF?
A64102N
B56111N
C4155Y
D3583N
E2354N
F1236Y

In the case of tied scores, the public vote wields the most power. In Prof Biggs’s example, Couple C were third with the judges, but last with the viewers. Their final total of five tied them with Couple E. However, Couple E’s higher score from the viewers saved them and put Couple C in the dance-off.

Prof Biggs added: “Under this system it is impossible for anyone to be safe on the judges' votes alone. In the extreme case where the judges’ list is 6, 5, 4, 3, 2, 1, the public list could be [the reverse ranking] 1, 2, 3, 4, 5, 6, resulting in final scores of 7, 7, 7, 7, 7, 7. Then the public vote decides.”

And in that scenario, the two acts placed highest with the judges go straight to the dance-off, because they have the lowest scores from the public. It would be one of the biggest shocks in Strictly history, but not mathematically impossible.

It's tough at the top

“It is always possible for the public vote to save the judges’ weakest person from the dance-off,” Prof Biggs said. “This is more likely when there are ties, because the weakest person will then have more than one point from the judges. For example, with five contestants, the judges’ points could be 5, 5, 4, 4, 3. If the 3-couple gets four or five points from the public, there will surely be two others below them.”

After the judges have given their scores, there are 120 possible ways that the public could rank the five couples. Prof Biggs found that the couple placed last by the judges escaped the dance-off in 48 of them. However, those 120 possibilities are not all equally likely.

It’s not all down to maths, though. In most cases, it’s down to the judges to decide who stays and goes following the dance-off itself. Even if that ends in a stalemate, head judge Shirley Ballas, with her vast experience in the ballroom, has the final say on who’s back for another week.

Sums can get you so far. But most of the time, it’s how you do those moves that matters.