50% of Ivy League students get this basic math problem wrong. Here’s why.

There’s a blindingly simple math problem that’s been giving Ivy League students—some of the sharpest young minds on the planet—trouble for years:

“A bat and a ball, together, cost $1.10. The bat costs $1 more than the ball. How much does the ball cost?”

It’s an extremely basic matter of addition and subtraction, not algebraic geometry. So why do so many people get this math question wrong?

The problem with our fast-thinking intuition

If you’re like most people, when you hear the bat-and-ball setup, your brain instantly screams at you that the answer is obviously 10 cents. The bat costs a dollar, and the ball costs 10 cents, totaling $1.10.

However, if you reread the question carefully, 10 cents is incorrect.

The ball actually costs 5 cents. The bat, which costs $1 more, would cost $1.05, giving a grand total of $1.10.

Psychologist Daniel Kahneman writes in his 2011 book Thinking, Fast and Slow that about 50% of students tested at elite schools such as the Massachusetts Institute of Technology, Princeton University, and Harvard University got the question wrong. He’s referring to a study conducted by Shane Frederick in 2005. Frederick administered a Cognitive Reflection Test (CRT) to students at these schools, featuring three tricky questions, including the bat-and-ball puzzle.

(All of the schools did poorly, but for bragging rights, MIT students did the best of the bunch.)

Kahneman explains that our urge to believe a quick, sensible answer that doesn’t require much thinking beyond the surface is powerful.

He writes in Heuristics and Biases, “Almost everyone we ask reports an initial tendency to answer ’10 cents’. … People are not accustomed to thinking hard, and are often content to trust a plausible judgment that quickly comes to mind.”

Here’s Kahneman explaining the concept in his own words:

Kahneman used this phenomenon to develop his idea of System 1 and System 2 thinking

In his book, he explains that System 1 thinking is automatic, fast, and intuitive. It takes no effort whatsoever. Answers appear to us almost instantly.

For example, we don’t have to think about the drive to work. We know the route and don’t consciously need to think about where or when to turn.

System 2 thinking is more deliberate and conscious. It requires real effort and reflection and is used to solve complex analytical tasks.

If the road we take to work is closed, for example, and we need to find an alternate route, this is the thinking system that kicks in.

The importance of Kahneman’s work lies in helping people understand that System 1 thinking is extremely powerful and stubborn. It’s not well monitored by System 2, and while it’s useful, it can often lead us astray.

Further research has shown that our fast thinking is extraordinarily stubborn

A 2023 paper in the journal Cognition presented a whopping 59 new studies on the bat-and-ball problem, covering more than 70,000 participants.

The findings were truly remarkable.

In five separate studies, the authors administered the bat-and-ball problem to thousands of volunteers. This time, however, they affixed one of several warnings to the problem. The warnings, marked with a large yellow exclamation point, said things like, “Be careful! Many people miss the following problem because they do not take the time to check their answer.”

“The warnings improved performance,” the authors write, “but not by much.”

In another variation of the test, they specifically told respondents that 10 cents was not the answer. In fact, they tested this eight times on nearly 8,000 people.

More people got the right answer in these tests (phew!), but many people still got the question wrong. The researchers argued that the “intuitive group” (people who go with a gut answer of 10 cents) should be further divided into:

• The careless: Respondents who did not put in the effort to think through the problem and revised their answers when given the chance.
• The hopeless: Respondents who were not capable of computing the correct answer.

The takeaways are fascinating: Two-system thinking may very well exist, but the solution is not just to “slow down and think harder.” Even when warned, prompted, and flat-out given the answer to this puzzle, many respondents continued to get it wrong.

It shows that once we’ve made up our minds about something, it’s extremely difficult for us to revise our judgment.

Not all of us are Ivy League students, but we’d all like to make better decisions. What can all of this research teach us about working with our brains instead of against them?

Slowing down to think through problems carefully can help, but research clearly shows that it’s not a foolproof solution.

What’s more important is how we frame the problems we face and prevent ourselves from getting locked into one way of seeing things. By then, it may be too late to change our minds.

Asking hypotheticals like, “What if the truth is actually the opposite of what I think?” and looking for evidence that disproves our theories is a good way to stress-test our ideas. Seeking out alternative perspectives before we’ve planted our flag in an answer or opinion can help ensure we’re still open and willing to change our stance.

It’s hard to overcome our brains’ built-in, natural biases. But even being aware of them can help us make better decisions throughout the day.