Try solving each expression before reading the explanation.
Puzzle 1: 6 + 4 × 2
Some people might calculate (6 + 4 = 10) and then multiply by 2, obtaining 20.
But multiplication comes first:
(4 \times 2 = 8)
Then:
(6 + 8 = 14)
Correct answer: 14.
Puzzle 2: 15 − 3 × 4
The tempting approach is to subtract first:
(15 - 3 = 12)
Then multiply by 4 to get 48.
However, the correct calculation begins with multiplication:
(3 \times 4 = 12)
Then:
(15 - 12 = 3)
Correct answer: 3.
Puzzle 3: 20 ÷ 5 + 3
Division comes before addition:
(20 \div 5 = 4)
Then:
(4 + 3 = 7)
Correct answer: 7.
Puzzle 4: 18 − 6 ÷ 2
First, divide:
(6 \div 2 = 3)
Then subtract:
(18 - 3 = 15)
Correct answer: 15.
Puzzle 5: 7 + 8 − 3
There is no multiplication, division, exponent, or parenthesis to consider. Addition and subtraction have equal precedence, so work from left to right:
(7 + 8 = 15)
Then:
(15 - 3 = 12)
Correct answer: 12.
These examples demonstrate that a small change in the operations can alter the calculation significantly. Learning to identify the highest-priority operation is the most important first step.
How Parentheses Can Change the Answer
Parentheses are another important part of arithmetic because they explicitly indicate which calculation should be performed first.
Compare these two expressions:
(8 - 2 \times 3 + 6)
and
((8 - 2) \times 3 + 6)
The first expression has a correct answer of 8 because multiplication takes priority over subtraction and addition.
The second expression is different because the parentheses require us to calculate (8 - 2) first:
(8 - 2 = 6)
Then multiply:
(6 \times 3 = 18)
Finally, add 6:
(18 + 6 = 24)
Therefore:
[
(8 - 2) \times 3 + 6 = 24
]
This is an important observation. The number 24 is not inherently an impossible result for a similar calculation. It is simply the wrong answer for the original expression because the original expression does not contain those parentheses.
Mathematical symbols matter. Even a small change in notation can change the meaning of an entire expression.
Why These Puzzles Are Useful for the Brain
Simple arithmetic puzzles can be useful mental exercises because they encourage people to slow down, pay attention to details, and question their first impressions.
When people encounter a familiar-looking problem, they may immediately begin calculating without checking the rules. A puzzle like this reveals how easily assumptions can influence the result.
Working through the expression step by step encourages several useful habits:
-
Paying attention to mathematical symbols.
-
Identifying which operation has priority.
-
Checking each step instead of rushing.
-
Recognizing and correcting mistakes.
-
Explaining a solution clearly to someone else.
These are valuable habits in mathematics and in many everyday situations that require careful reasoning.
However, solving a single arithmetic puzzle does not provide a reliable measure of intelligence. Someone who answers incorrectly may simply have forgotten a rule or misread the expression. Mathematical accuracy improves through practice, explanation, and repetition.
The goal should be to understand the reasoning rather than judge people by whether they immediately produce the correct number.
A Common Mistake: Trusting the First Answer
One reason puzzles like this attract attention is that the incorrect answer can feel convincing.
When someone calculates 24, the individual steps may appear reasonable. Subtracting, multiplying, and adding are all familiar operations. The problem is not necessarily a lack of mathematical ability; it is the decision to apply the operations in the wrong order.
This illustrates an important lesson about problem-solving: an answer can look logical while still being incorrect.
A good habit is to pause before accepting the first result. Ask yourself whether you have followed the relevant rules and whether each step preserves the meaning of the original expression.
This approach is useful beyond arithmetic. In everyday life, an initial impression can be misleading when important information has been overlooked. Taking time to examine the details can prevent avoidable errors.
How to Teach the Rule to Children
This puzzle can also be used to help children understand why arithmetic expressions must be evaluated in a consistent order.
Rather than simply telling them to memorize PEMDAS, explain what the rule accomplishes.
Start with a simple expression such as:
(2 + 3 \times 4)
Ask the child to calculate the answer. Some may obtain 20 by adding first, while others may obtain 14 by multiplying first.
Then explain that mathematics uses a shared convention: multiplication is performed before addition unless parentheses indicate otherwise.
The correct calculation is:
(3 \times 4 = 12)
Then:
(2 + 12 = 14)
Afterward, introduce examples involving subtraction, division, and parentheses.
Encourage children to explain each step in their own words. Understanding why an operation comes first is more valuable than memorizing a sequence of letters without knowing what they mean.
It is also important to treat mistakes as opportunities to learn. A student who understands why an answer is incorrect is making progress, even if the first attempt was unsuccessful.
Final Answer: 8, Not 24
The puzzle in the image asks us to solve:
[
8 - 2 \times 3 + 6
]
The correct answer is 8.
The multiplication must be completed first:
(2 \times 3 = 6)
The expression then becomes:
(8 - 6 + 6)
Subtraction and addition have equal precedence, so we work from left to right:
(8 - 6 = 2)
Finally:
(2 + 6 = 8)
The answer of 24 results from incorrectly performing the subtraction before the multiplication.
The central lesson is straightforward: always follow the order of operations, and do not automatically calculate from left to right when multiplication, division, addition, or subtraction have different priorities.
The next time you encounter a math puzzle that seems almost too simple, take a moment to examine the symbols and identify which operation should come first.
Sometimes, the difference between a wrong answer and a correct one is not complicated mathematics at all. It is simply knowing the rule—and remembering to use it.