The Bat and Ball Problem: Why Most People Get It Wrong

A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

If an answer popped into your head straight away, it was probably 10 cents. It pops into almost everyone’s. It feels obvious, almost too easy to be a puzzle. And it’s wrong.

Checking the 10-cent answer

Put the answer back into the question. If the ball costs 10 cents, the bat costs $1.00 more, which is $1.10. Together that’s $1.20, not $1.10.

The right answer is 5 cents. The ball is 5 cents, the bat is $1.05, and together they make $1.10. The bat costs exactly $1.00 more than the ball. Slide the ball’s price in the toy at the top and watch the total: you can feel how hard the 10-cent answer pulls, even when the bar shows it doesn’t fit.

If you like algebra, the working is short. Call the ball x. The bat is x + $1.00. Together, x + x + $1.00 = $1.10, so 2x = $0.10, and x = $0.05.

Nothing about this needs advanced maths. The puzzle is hard for a different reason, and that reason is the interesting part.

Smart people get it wrong too

The problem was made famous by the psychologist Shane Frederick, who in 2005 included it in a three-question test he called the Cognitive Reflection Test. He gave it to thousands of people, many of them students at leading universities.

More than half of the students at Harvard, MIT and Princeton gave the 10-cent answer. At some other universities, a large majority did. This isn’t because they couldn’t do the sum. If you ask anyone to work out 5 cents plus $1.05, they get $1.10. It’s because the sum never got done. The answer arrived first, felt right, and was handed in.

Frederick’s other two questions work the same way.

Each has an answer that arrives instantly and fits a pattern: split $1.10 into $1 and 10 cents; match 5, 5, 5 with 100, 100, 100; halve 48. Each takes only a moment of checking to see through. And each catches a large share of very capable people.

Fast thinking and slow thinking

Psychologists often describe two modes of thinking, an idea Daniel Kahneman made famous in Thinking, Fast and Slow:

• Fast thinking (sometimes called System 1) is automatic, effortless and immediate. It’s what reads a face, finishes a familiar sentence or tells you 2 + 2 is 4.
• Slow thinking (System 2) is deliberate and effortful. It’s what multiplies 17 by 24, checks a claim or works through a problem step by step.

Fast thinking is right most of the time, which is why we rely on it. The bat and ball problem is built so that fast thinking produces a confident wrong answer, and slow thinking has to step in to check it. Most people don’t notice there’s anything to check, because the answer feels so right.

It’s worth being precise about what the problem shows. It isn’t that intuition is bad. Intuition is how experts make fast, excellent decisions in their own field: a chess grandmaster sees the right move, a doctor spots a sick child across a room. It’s that intuition can be confidently wrong in situations it hasn’t been trained on, and it doesn’t come with a warning light.

The mind may notice anyway

There’s an intriguing twist. In 2013, the psychologist Wim De Neys and colleagues compared people answering the bat and ball problem with people answering a version without the trap, in which the bat simply costs $1 and the ball some amount. People who gave the wrong 10-cent answer on the tricky version were less confident in it than people answering the easy version were in theirs.

In other words, part of the mind seems to register that something is off, even when the person goes with the quick answer anyway. That’s hopeful. It suggests that the habit to build isn’t a new kind of intelligence, but paying attention to a small feeling of doubt that’s already there.

Why it matters beyond puzzles

Frederick found that people who did better on his three questions also made different decisions about money and risk. For example, they were more willing to wait for a larger payment later rather than take a smaller one now, and more willing to take a gamble with a higher expected value. Other researchers have found that cognitive reflection predicts less belief in some kinds of misinformation and more accurate judgements in some tasks.

The connection isn’t that people who solve the puzzle are cleverer. It’s that they’re in the habit of checking answers that come too easily. That habit is useful whenever a situation offers a tempting quick answer: a price that looks like a bargain, a headline that confirms what you already think, a statistic that seems to settle an argument.

More traps with the same shape

Once you know the trick, you start to see it in other famous questions. They all offer an answer that fits the surface of the question while ignoring a detail.

How many animals of each kind did Moses take on the ark? Most people say two. It was Noah who built the ark, not Moses. In a 1981 study, Thomas Erickson and Mark Mattson found that most people answer without noticing the wrong name, even when warned that some questions contain errors. It’s called the Moses illusion, and it works because Moses is close enough to Noah to pass.

A farmer has 17 sheep. All but 9 die. How many are left? The quick answer is 8, from subtracting 9 from 17. But “all but 9 die” means 9 survive.

If you’re running a race and pass the person in second place, what place are you in? First feels right. You’re now second: you took their place, and the leader is still ahead.

None of these needs knowledge. They need a pause, and a second look at the exact words. That pause is cheap, and it’s the whole skill the bat and ball problem is measuring.

How to catch yourself

You can’t switch off fast thinking, and you wouldn’t want to. But you can learn where it’s likely to trip you up, and build a short habit.

1. Notice when an answer comes too easily on something that’s meant to be tricky. Puzzles, exam questions and sales pitches are often designed so that the obvious answer is the trap.
2. Check the answer against the question. Put it back in. If the ball were 10 cents, would the total be $1.10?
3. Slow down for numbers. Sums, percentages and comparisons are where quick answers most often go wrong.
4. Explain your answer in a sentence. If you can’t say why it’s right, it might not be.
5. Ask what it would look like if you were wrong. It’s the same question that protects against confirmation bias, and it works here too.

Has the problem lost its bite?

Because the bat and ball problem is now famous, many people know the answer before they read the question, which makes it less useful in research. Psychologists have built new versions with the same structure, such as Toplak, West and Stanovich’s expanded test in 2014. The new questions still catch a large share of people, which suggests the effect is real and not just a quirk of one wording. Being told about the trap helps with that trap. It helps much less with the next one.

Test yourself

The bat and ball is the first question in the free thinking traps test, followed by nineteen more puzzles, biases and bad arguments. For the bigger family of mental shortcuts this belongs to, see cognitive biases that fool smart people.