Why this divisive math problem left everyone thinking they know best

Last week a seemingly simple math problem caused deep division among DailyMail.com readers, igniting an argument so furious that it resembled an angry political debate.

A British mathematician and TV host, Carol Vorderman, was irate enough to take to Twitter to call it an ‘attempt to alter the truth/rules of algebra’ and demand an apology to all math teachers. 

The story’s most popular comment said the answer is ‘Absolutely definitely 1 and I will fight anyone who says otherwise’.

Dailymail.com spoke to math experts to understand how algebra rage could become a topic so polarizing that people were willing to fight over it. Most importantly, we wanted to find out once and for all who is right.

Unfortunately, you might not be satisfied with the answer – which is that there is no ‘objectively’ correct solution. That’s because, while the principles of math aren’t subject to interpretation, the symbols invented to describe it are.

‘It’s a very artfully contrived problem. It’s like it’s designed to maximize confusion,’ said Steven Strogatz, a math professor at Cornell University who has written about this type of equation before. ‘You have to admire the cunning of whatever little devil created them,’ he added. 

The problem from last week was posed as above. 

Many people were convinced that the only truthful answer is 1. However, some computers and calculators produced an answer of 9. 

‘It’s not as cut and dried as [one] thinks. I might have thought that at first… but when you really dive into it there’s a lot going on,’ said Strogatz.

Many simplified 6 ÷ 2(1 + 2) to 6 ÷ 2(3).

It was undisputed that the first operation to deal with was the addition of 1 and 2

But the order in which operations should be applied from here is not objective. Therefore everything came down to what part of the equation you did first: expanding the bracket or carrying out the division.

PEMDAS (or BIDMAS/BODMAS as it is often taught in the UK) is the simplistic formula provided to students in primary school.

Strict and pedantic application of PEMDAS says calculations should be made from left to right because division and multiplication are of equal priority. That makes the answer 9.

Expressing the original problem in this less elegant fashion and pedantically applying PEMDAS yields an answer of 9

However, many readers applied a more sophisticated convention that dictates ‘implicit’ multiplication is prioritized over ‘explicit’ multiplication or division.

This means when multiplication is implied by a bracket – in this case 2(3) – it is processed before multiplication or division – which is signaled explicitly with an ‘x’ or ‘÷’ symbol.

‘The whole issue boils down to, does implicit multiplication get higher priority than explicit division? With the 6 divided by 2 symbol, there’s an explicit division symbol, but there’s only an implicit multiplication symbol,’ said Strogatz. 

This approach is clearly indicated when the problem is expressed in a form that excludes the division symbol, and uses fractions instead:

Expressing the original problem in this notation makes the solution 1 appear more intuitive. The division symbol (÷) is very rarely used in high school math and beyond

Last week’s problem elicited a furious reaction from many, including British TV personality Carol Vorderman (pictured) who said the answer was clearly 1

Steven Strogatz, a professor of applied mathematics at Cornell University, described the class of problems as ‘artfully contrived’

What makes the original problem especially confusing is that it combines basic notation (the division symbol) with implicit multiplication (the brackets). 

Further evidence that the interpretation of the original problem is not black and white lies in the fact that calculators and other computational ‘answer engines’ deal with it differently.

Wolfram Alpha, one of the most established semantic search engines, as well as Google, give the answer as 9. They are not wrong because they also specify how they chose to interpret the confusing division symbol.

Strogatz pointed out another programming language, Julia, that is capable of applying the second approach which considers implicit multiplication.

High school math students may recall that electronic calculators would solve most problems accurately, but that for PEMDAS-type questions they would often apply operations in an order that would produce answers teachers consider incorrect.

In fact, the Physical Review Letters, one of the most recognized academic journals in Physics, provides editorial direction to avoid dilemmas exactly like this.

Strogatz pointed to that need to clear up any misunderstanding as proof there is no one universally accepted convention. 

‘People need to be told,’ he said.

Presh Talwalkar, who created the YouTube video from which last week’s puzzle came, agreed that the solution is not black and white.

‘We should remember this question actually went viral years ago because two calculators gave different answers. If you group, the answer is 1. If you use PEMDAS, the answer is 9,’ he said. 

‘In general, mathematical notation can have multiple meanings,’ he said. ‘So I think it’s important to know the context of the question. I completed a math degree, and I can honestly say we never had any arguments about ambiguous expressions. Most textbooks are precise in defining a convention or using parentheses carefully.’

Strogatz added that, although the controversy surrounding the problem is divisive, it is not ultimately an issue of math.

‘It’s kind of interesting, more sociologically than mathematically or pedagogically. There’s a lot of different things that play here,’ he said.