Colin Beveridge

Maths, clearly

This story is based around a genuine situation. I have deliberately not mentioned the candidate or party involved by name, but it would hardly take Rebus to work it out.

“Miserable out.”

Nick Bourbaki didn’t open his eyes, let alone take his steel-capped size-nines off of his replica oak desk. The voice had knocked. Trouble didn’t knock.

“Lucky I’m indoors.” In his head, a snide remark about not moving here for the weather.

“I brought you something,” said the voice. A very slight cardboard thud. A splash. A smell. Not good coffee, but plenty of it.

Bourbaki sighed, opened a languid eye and sighed again. He recognised the face attached to the voice, from the local paper, from election leaflets, from never-ending adverts online. “I don’t do politics. Not since the incident with the Lib Dem graphs.”

“I’m not a Lib Dem,” the face said with a wry smile. “We’re different.”

They said they were different. You’re all the same.”

“I have something that might interest you,” said the face, patiently.

Fiona Drummond moved her martini glass to one side to get a better view. It contained only lemon-soda. It’s never too early to keep up appearances. “So what is it?”

“Election results,” said Bourbaki.

“Oh!” said Drummond, quickly Sharpie-ing an arrow with “CAN’T WIN HERE!” next to the winner’s tally. “So… this guy with 50% won?”

“Yes, but not at first. It’s an STV election.”

Big Joe Tracy made a face, and then took a sip of his decaf pumpkin spice mochachino with extra whipped cream. Wrong way round, thought Bourbaki.

Drummond cocked an eyebrow. “What’s STV?”

“It stands for Second-most Terrible Vote.”

Bourbaki sniffed. “It’s better than first past the post, although here it gave the same result. Let me run it through.”

CandidateRound 1TransfersRound 2

“This election had four candidates, and voters can rank them in order.” This was a non-smoking establishment, and Bourbaki didn’t smoke. But it still felt like a moment to light a cigarette.

“Wait,” said Drummond. “The numbers don’t add up to 100%!”

“Rounding error,” drawled Tracy, as if talking to a five-year-old he didn’t much care for.

“Well, yes and no,” said Bourbaki. “In the sense of ‘when you round numbers, you can wind up with columns that don’t add up’, that’s not what’s happened. They’ve incorrectly rounded some of the numbers.”

Four eyebrows raised so far and so quickly that the ceiling fan looked, for a moment, to be in some danger.

“I know, shocking,” said Bourbaki. “This is why nobody trusts the polls. But yes: some of these numbers are slightly off in the last decimal place.”

“Can we go back to the stupid system now?” Tracy took another slurp of his so-called coffee. “Four candidates. Voters rank them in order.”

“And – as you can see – the count consists of one or more rounds. In each round, each ballot counts as a vote for the best-ranked candidate still in the election.”

“Right.” Drummond pointed with the end of the cocktail stick that didn’t have an olive. “So in the first round, each ballot counts for the voter’s first preference. There’s the two front-runners with 50% and 42%, then two also-rans on about 5% and 3%.”

“To win the election, you need to have strictly more than 50% of the remaining votes.”

“Oo, so close!”

“Close indeed – candidate 2 was one vote short of winning in the first round.”

“So what happens next? All the second choices are counted?”

“No – after that, the least-preferred candidate is eliminated, and their votes redistributed.”

“So, candidate 1 is in last place at the end of round 1, so their votes now count for the second choice candidate on those ballots?”

“If there’s a second preference, yes – here, about 12% of candidate 1’s voters didn’t express one. They’re the non-transferable ones.”

“And after the second-preference votes are reallocated to the remaining candidates, the winner now has more than 50% of the votes and wins.”

“Bingo. Question is, how many votes were cast?”

“Not enough information,” said Drummond and Tracy at once.

“There are fewer than 1,000 voters,” said Bourbaki. “There’s no solution if the numbers are consistently rounded – but if they’re occasionally truncated, there is.”

Can you work out how many votes were cast?

“My first thought was continued fractions,” said Bourbaki, “but it gets… messy. There’s too big a gap to make it work.”

“You could just write code,” said Tracy.

“That’s your answer for everything.” Another response in harmony.

“Wait,” said Drummond.

They waited.

“This second column… that’s all of candidate 1’s votes. 2.89% of the total. Certainly less than 30 votes.”

Bourbaki and Tracy knew better than to interrupt.

“11.76% is less than an eighth, but more than a ninth. There’s not many numbers that can work. It can only be two seventeenths, three twenty-fifths or three twenty-sixths.”

“And it’s not three twenty-fifths,” said Tracy.

“Right. Two seventeenths is twelve hundred-and-twoths…” (A brief, you-heard-me glare) “… twelve less 2% is 11.76! Just to check, three twenty-sixths is twelve hundred-and-fourths, which is about 11.5.”

“If you know your ninja tricks…”

“Shut it, Tracy. Candidate 1 got 17 votes.” Drummond grinned.

Tracy, in time-honoured tradition, took a Sharpie to a napkin. “And 17 is 2.89% of 588.”

Bourbaki unfolded his piece of paper with a smile. “Personally, I just called the polling company. The winner had 294 votes in the first round, followed by 248, 29 and (as you surmised), 17. Of those 17, two were non-transferable, three went to the second-placed candidate, five to the winner and seven to the third.”

“Was the candidate happy?”

“You can never tell with politicians. He bought the coffee, though.”