Puzzles, arguments and the wider world

You know the rule, why it works and where it came from. This layer takes it out into the wild: viral puzzles that start arguments online, number games that people have played for over a century, olympiad problems, how the rule lives inside spreadsheets and programming languages, projects you can run yourself, and questions that are still genuinely open.

Some of these puzzles have no single right answer. That is the point: part of being good at maths is recognising when a question is badly posed, and saying so clearly.

Every few months a puzzle like this goes viral:

8 ÷ 2(2 + 2) = ?

Half the comments say 16. The other half say 1. Both sides are sure, and both sides think the other side failed school maths. Who is right?

The "16" reading. The bracket is 4. The 2(4) just means 2 × 4, an ordinary multiplication. So the expression is 8 ÷ 2 × 4, and ÷ and × go left to right: 8 ÷ 2 × (2 + 2) = 8 ÷ 2 × 4 = 4 × 4 = 16.

The "1" reading. Writing 2(2 + 2) with no × sign, called implied multiplication or multiplication by juxtaposition (putting things side by side), makes 2(2 + 2) a single tight unit, like a term in algebra. So the expression is 8 ÷ [2 × (2 + 2)] = 8 ÷ 8 = 1.

The honest answer: the expression is badly written. The basic school rule, applied strictly, gives 16. But many mathematicians, scientists and some calculators treat implied multiplication as binding more tightly, which gives 1. When a well-meaning expert could reasonably read it two ways, the fault is in the writing, not the reader.

A puzzle that has been played for well over a hundred years: make every whole number you can using exactly four 4s, with + − × ÷ and brackets. You may also join 4s into 44 or 444.

Here are 0 to 10:

Now try 11. And 13. You will struggle, and not because you are missing a trick.

Fix the rules exactly: exactly four 4s, joined into 4, 44, 444 or 4444 if you wish; only + − × ÷ and brackets; no decimal point, no √, no factorial, and no minus sign in front of a number on its own. Under those rules a computer can check every possible arrangement — every way of splitting the 4s, every order, every bracketing — and the numbers from 0 to 20 that cannot be made are exactly 11, 13, 14, 18, 19.

Change the rules and the list changes: if you are not allowed to join 4s at all, 10 becomes impossible too. That is the real lesson — an "impossible" claim in a puzzle only means anything once the allowed moves are written down.

To reach the missing numbers, puzzle fans allow extra tools:

• √4 = 2 (square root): 13 = 44 ÷ 4 + √4 = 11 + 2.
• .4 = 0.4 (a decimal): 11 = 4 ÷ .4 + 4 ÷ 4 = 10 + 1; 18 = 44 × .4 + .4 = 17.6 + 0.4.
• 4! = 4 × 3 × 2 × 1 = 24 (factorial): 19 = 4! − 4 − 4 ÷ 4 = 24 − 4 − 1.
• 14 = 4 × (√4 + √4) − √4 = 16 − 2.

Every one of these still relies on the order of operations to mean exactly one thing.

The 24 game is played with four numbers (often from playing cards, 1 to 13). Use each number exactly once, with + − × ÷ and brackets, to make exactly 24. Why 24? Because it has so many factors (1, 2, 3, 4, 6, 8, 12, 24) that most sets of four numbers can reach it.

The key strategy is working backwards: 24 = 3 × 8 = 4 × 6 = 2 × 12 = 1 × 24, or 20 + 4, or 30 − 6. Look at your numbers and ask which of these you can build.

• 4, 7, 7, 7: 7 ÷ 7 = 1, 7 − 1 = 6, 4 × 6 = 24. So 4 × (7 − 7 ÷ 7) = 4 × (7 − 1) = 4 × 6 = 24.
• 1, 2, 3, 4: 1 × 2 × 3 × 4 = 2 × 3 × 4 = 6 × 4 = 24.
• 2, 3, 4, 9: a tempting try is (9 − 3 − 2) × 4 = (6 − 2) × 4 = 4 × 4 = 16, which misses. Work backwards instead: 24 = 48 ÷ 2, and 48 = 12 × 4 = (9 + 3) × 4. So (9 + 3) × 4 ÷ 2 = 12 × 4 ÷ 2 = 48 ÷ 2 = 24.

Olympiad problems on this topic rarely ask you to grind through a long expression. They ask you to see structure: pair terms, spot a zero, count cleverly, or search systematically.

Every programming language has an operator precedence table, its own written-down order of operations. For ordinary arithmetic they almost all agree with school maths: in Python, Java, C, JavaScript, Scratch and spreadsheets, 2 + 3 * 4 is 14.

But at the edges they can differ, and professional programmers learn to add brackets whenever there is the slightest doubt.

• Symbols. Computers use * for × and / for ÷. Powers are ** in Python and ^ in spreadsheets.
• Whole-number division. In Python, 7 / 2 is 3.5, but 7 // 2 is 3 (the quotient) and 7 % 2 is 1 (the remainder). // and % share a level with * and /.
• Negative squares. In maths, −3² means −(3²) = −9. Python agrees: -3**2 is −9. But Microsoft Excel gives 9 for =-3^2: its published precedence table ranks negation above exponentiation, so the minus is applied first. Spreadsheets that follow Excel's rules behave the same way.
• Stacked powers. In maths, 2^3^2 means 2^(3^2) = 2⁹ = 512, and Python agrees. Excel treats ^ like any other equal-rank operator and works left to right, so =2^3^2 is (2³)² = 64.

Who uses this every day?

• Accountants and shopkeepers build spreadsheets of bills, taxes and discounts. A missing bracket in a GST formula such as price × (1 + rate ÷ 100) can mis-price every item in a shop.
• Software engineers write expressions in code millions of times; the precedence table of their language is second nature.
• Engineers and scientists plug numbers into formulas like distance = speed × time + ½ × a × t². The order of operations decides what the formula means.
• Data analysts compute averages: (sum of values) ÷ (count). Forgetting the brackets and dividing only the last value is one of the most common spreadsheet errors.
• Teachers and textbook writers have to write expressions that cannot be misread, which is harder than it looks!

Our way of writing, with the operation between its numbers (infix), is the reason we need an order of operations at all. Other notations avoid the problem:

• Prefix (Polish) notation: the operation comes first. 2 + 3 × 4 is + 2 × 3 4. The programming language Lisp writes it as (+ 2 (* 3 4)).
• Postfix (Reverse Polish): the operation comes last. 2 3 4 × +. No brackets are ever needed.
• Fraction bars and stacked layouts: the position on the page shows the grouping.
• Expression trees: a picture of the structure, with the last operation at the top.

Why do we still use infix? Because it is compact and close to how we speak ("two plus three times four"), and once everyone knows the convention it is fast to read. It is a trade-off: a convenient notation that needs a shared rule.

A famous television numbers game (called Countdown in the UK, and copied in many countries) gives players six numbers and a three-digit target. Using + − × ÷ and brackets, each number at most once, and every step a whole number, they have 30 seconds to hit the target, or get as close as possible.

Small numbers are chosen from 1 to 10 and big numbers from 25, 50, 75 and 100. A typical round: 75, 4, 7, 2, 9, 1, target 307.

Work backwards: 307 is close to 300 = 4 × 75. The difference is 7, and we have a 7! So 4 × 75 + 7 = 300 + 7 = 307. Three numbers are left over, which is allowed.

The order of operations is what lets you write the answer as one short line: 4 × 75 + 7 needs no brackets, because the × is done first anyway.

A shop has a ₹1,000 shirt on sale. There are two things to apply: a ₹100 discount and 18% GST. Does the order matter?

• Discount first, then GST: (1,000 − 100) × 118 ÷ 100 = 900 × 1.18 = ₹1,062.
• GST first, then discount: 1,000 × 118 ÷ 100 − 100 = 1,180 − 100 = ₹1,080.

They differ by ₹18, which is exactly the 18% GST on the ₹100 discount. The brackets decide whether the tax is charged on the discounted price or on the full price. In India the first way is the usual one: the GST Council’s own guidance says that "pre-supply discounts i.e. discounts recorded in the invoice" are left out when working out the taxable value, so a discount printed on the bill is taken off before GST is added.

Now try a 10% discount instead of ₹100. Discount first: 1,000 × 90 ÷ 100 × 118 ÷ 100 = ₹1,062. GST first: 1,000 × 118 ÷ 100 × 90 ÷ 100 = ₹1,062. The same! Percentage changes are multiplications, and multiplications can be done in any order. A flat discount is a subtraction, and mixing it with a multiplication makes the order matter.

A gurdwara langar or a wedding caterer may cook for thousands of people. Big numbers make the order of operations even more important, because a slip is multiplied too.

A caterer plans for 1,200 guests. Each plate gets 2 rotis, 150 g of rice and 120 g of dal. The kitchen also cooks 10% extra for safety.

• Rotis: 1,200 × 2 × 110 ÷ 100 = 2,640.
• Rice in kg: 1,200 × 150 × 110 ÷ 100 ÷ 1,000 = 198 kg.
• Dal in kg: 1,200 × 120 × 110 ÷ 100 ÷ 1,000 = 792/5 kg.

Each is a long chain of × and ÷ done left to right. Multiplying first and dividing last keeps every step a whole number, which is useful when working by hand.

Here are four more problems of the kind that appear in maths olympiads and puzzle contests. Try each before reading the solution. The skills they reward are the ones from this whole topic: knowing exactly what an expression means, working systematically, and spotting structure.