A bat and a ball problem. Two modes of thinking: fast (intuitive) and slow (analytical)

Problem. A bat and a ball cost $1.10 together. The bat costs $1 more than the ball. How much is the ball?

Answer. The ball costs $0.05.

Fig. 1. A bat and a ball problem

Solution. While not difficult, this puzzle often leads to incorrect answers due to psychological factors. It gained fame through Daniel Kahneman, an expert in the psychology of judgment and decision-making, who featured it in his best-selling book "Thinking, Fast and Slow". The book examines human behavior by analyzing how our brain processes information. Kahneman explains that our brain operates in two modes: fast and slow. The fast mode is always active, permanently running in the background, to help us deal with routine tasks and respond swiftly in case of an emergency. The slow mode engages when we pause to evaluate the situation and find the best solution. Put it this way, the first mode can be described as intuitive, while the second is analytical.

The apparent simplicity of the puzzle triggers a fast response, resulting in the wrong conclusion. We see two numbers, $1.10 and $1, which can be easily subtracted, and we know we need to determine the difference between specific values. The brain picks up the easy clues and deduces that the ball must cost 10 cents. However, the ball is $1 cheaper than the bat, not their combined cost. This key detail is hidden from view by the clever choice of numbers and the puzzle's deceptive setup.

If you also fell into this trap, don't worry. You are in good company. A group of Harvard students made the same mistake when pressed for a prompt response, even though they could handle far superior conundrums. Our fast mode picks up superficial clues, which can mislead us into making wrong decisions. The puzzle was created by Yale professor Shane Frederick, an expert in consumer behavior, to illustrate how skillful wording can shape our perception of a product and affect our spending habits.

We are all guilty of buying things we don't need, swayed by tempting offers like 'Buy One Get One Free'. Another example of consumer manipulation involves pricing tags such as $59.99 instead of $60. Saving a single cent while spending tens of dollars can't improve our finances. Still, these strategies work, and we can't deny it. There is a science behind our psychological traits, which marketing professionals use to encourage impulsive buying and assure us we've made the right choice.

I will rephrase the puzzle to correct the misleading wording, and the solution will become clear:

A bat and a ball cost $1.10 together.

The difference between their prices is $1.00.

How much is the ball?

We now see we have two unknowns and two equations to solve. I will employ visual object recognition to streamline the solution. This tactic is commonly used in consumer marketing and educational strategies, albeit for different purposes. Your education helps you withstand manipulations and form independent opinions that benefit you and not others. The two equations are as follows:

The correct answer is that the ball costs $0.05, and the bat costs $1.05.