8 ÷ 2(2+2): Is the Answer 1 or 16?

Short answer: with the standard order of operations taught in schools and used by most calculators and programming languages, $8\div 2(2+2)=\mathbf{16}$. The answer 1 comes from a different, older convention in which “$2(2+2)$” is treated as a single block. The real problem is the notation: the expression is ambiguous, and nobody writing math seriously would write it this way.

Contents7 sections

How you get 16

Parentheses first, then multiplication and division at the same level, from left to right:

$$8\div 2(2+2)=8\div 2\times 4=4\times 4=16.$$

Writing $2(4)$ instead of $2\times 4$ does not change anything in this reading: it is still a multiplication, and it waits its turn behind the division on its left.

How you get 1

Many people, and some textbooks, treat implied multiplication (a number written right next to a parenthesis, as in $2(4)$, or next to a letter, as in $2x$) as binding more tightly than $\div$. Then $2(2+2)$ is one quantity:

$$8\div 2(2+2)=8\div[2(2+2)]=8\div 8=1.$$

This is not as strange as it looks. In algebra almost everyone reads $1/2x$ as $\frac{1}{2x}$, not $\frac{1}{2}x$. People who learned it that way see $2(2+2)$ the same way.

So which one is right?

Both readings follow a rule; they just follow different rules. The most widely taught rule today (PEMDAS in the US, BODMAS or BIDMAS elsewhere, read correctly as “multiplication and division together, left to right”) gives 16. The “implied multiplication first” convention gives 1. Calculators disagree too: models that give implied multiplication a higher priority return 1, others return 16.

Mathematicians avoid the whole question by never writing $\div$ next to implied multiplication. They write a fraction instead, which makes the intent visible:

$$\frac{8}{2}(2+2)=16\qquad\text{or}\qquad \frac{8}{2(2+2)}=1.$$

Why it went viral

This exact expression spread on Twitter in late July 2019 and split replies between 1 and 16, and almost the same puzzle had circulated in 2011 and 2016. It works as bait because both camps can show a rule they were taught, so neither side can convince the other.

Try these

Puzzle 1

What is $6\div 2(1+2)$ with the standard left-to-right rule, and with “implied multiplication first”?

Show Answer

Standard: $6\div 2\times 3=3\times 3=\mathbf{9}$. Implied multiplication first: $6\div 6=\mathbf{1}$. It is the same trap as $8\div 2(2+2)$.

Puzzle 2

Is $12\div 3\times 2$ ambiguous?

Show Answer

No. There is no implied multiplication, and every convention agrees on left to right: $12\div 3=4$, then $4\times 2=\mathbf{8}$. Doing the multiplication first (getting 2) is simply a mistake.

Puzzle 3

Rewrite $8\div 2(2+2)$ so that everyone gets 16, and again so that everyone gets 1.

Show Answer

For 16: $(8\div 2)(2+2)$ or $\frac{8}{2}\times(2+2)$. For 1: $8\div(2(2+2))$ or $\frac{8}{2(2+2)}$.

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