The simple maths puzzle that shows us how to separate fact from fiction

30th November, 2023

For certain kinds of questions, there are answers that are simple, elegant and wrong. Take the most famous example of the genre, the “bat and ball” question: if a bat and a ball together cost $1.10, and the bat costs a dollar more than the ball, how much does the ball cost?

This is known as a cognitive reflection problem, because it’s designed to be a test of your ability to stop and think rather than a test of sophisticated maths. There’s a tempting wrong answer: 10 cents. But a moment’s reflection says that can’t be right: if the ball costs 10 cents, then the bat costs $1.10 and the two together don’t cost $1.10. Something doesn’t add up.

The bat and ball problem was developed by the behavioural economist Shane Frederick of Yale University and made famous by Nobel laureate Daniel Kahneman in his book Thinking, Fast and Slow. It’s an elegant illustration of Kahneman’s model of the human mind, which is that we have two modes of thinking. There’s a fast, intuitive processing system, which solves many problems with graceful ease but can also be lured into error, and there’s a slower, more effortful logic module, which can grind out the right answer when it must.

Frederick’s bat and ball problem offers an obvious decoy for the fast-thinking system to grab, while also having a correct answer that can be worked out using simple algebra or even trial and error. Most people consider the decoy answer of 10 cents even if they eventually produce the correct answer. The decoy answer is more popular when people are distracted or rushed and the correct answer takes longer to produce. (Have you got it yet?)

Frederick’s poser is not merely a curiosity: research by the Cornell psychologist Gordon Pennycook and others has found that people who score well on problems such as the bat and ball do a better job of distinguishing truth from partisan fake news.

The problem also raises some intriguing questions about the dual-system model of the mind. For example, when people get the answer wrong, what intuitive shortcut is leading them astray? And are they really wrong because they are careless? Or is it because the puzzle is beyond their capabilities?

In a fascinating new article in the journal Cognition, Andrew Meyer and Shane Frederick unleash a barrage of new studies, many of them subtle tweaks of the bat and ball problem. These tweaks enable Frederick and Meyer to distinguish between people who err because they subtly misread the question and those who thoughtlessly subtract the smaller number from the larger one. The truth is murkier than the fast- and slow-thinking model: there are different intuitions and different ways to be wrong.

I suppose that shouldn’t be a surprise. Pennycook reminds me that “the bat and ball question is just a single problem and if you think about the way we think in the real world, it’s obvious that our intuitions are varied and complicated”.

What blew my mind about Meyer and Frederick’s article was the way they painstakingly undermined the idea that made the bat and ball question famous — which is that many people can figure out the right answer if only they slow down for long enough to avoid the decoy. Meyer and Frederick suggest that this is not the case. They try variants on the question: in one case people are told, “HINT: 10 cents is not the answer”; in another they are offered the bold prompt, “Before responding, consider whether the answer could be five cents”. Both prompts help people find the right answer — which is, yes, five cents — but in many cases, people still don’t figure it out.

Some experimental subjects were given the question, followed by the bold and explicit statement: “The answer is five cents. Please enter the number five in the blank below: ___ cents.” More than 20 per cent of people did not give the correct answer despite being told exactly what they should write. Are they just not paying attention at all? Surely not.

“They definitely ARE paying attention,” Frederick tells me in an email. More likely, he says, they are stubbornly clinging to their intuitive first guess and are fearful of being tricked by a malevolent experimenter.

Pennycook agrees. “There’s always 20 per cent,” he offers, somewhat tongue in cheek. “Twenty per cent of people have crazy beliefs, 20 per cent of people are highly authoritarian.” And 20 per cent of people will not write down the right answer to a maths problem even when it’s handed to them on a plate, because they trust their gut more than they trust some tricksy experimenter.

Meyer and Frederick propose that we could sort the responses to the bat and ball question into three buckets: the reflective (taking the time to get it right the first time), the careless (who succeed only when given a prompt to think harder) and the hopeless (who cannot solve the problem even with heavy hints).

If this was just about funny logic puzzles, it would all be good clean fun. But the stakes are higher: remember Pennycook drew a clear connection between the ability to solve such puzzles and the ability to spot fake news. I argued in my book How to Make the World Add Up that a few simple mental tools would help everyone think more clearly about the numbers that swirl around us. If we calmed down, slowed down, looked for helpful comparisons and asked a couple of basic questions, we’d get to the truth.

I didn’t have the vocabulary at the time, but implicitly I was arguing that we were careless, not hopeless. I hope I was right. After some reflection, I am not so sure.

Written for and first published in the Financial Times on 3 November 2023.

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