Tricky math questions that catch people out

Maths questions split a pub quiz room like nothing else. Half the teams groan, the other half grab a beer mat and start scribbling. Here are 20 tricky maths questions in four rounds, from order of operations to probability puzzles that fool people with degrees. Every one has a single right answer and a one-line working, so you can settle arguments on the spot. We've also fixed the famous viral sums, which turn out to be genuinely ambiguous (more on that in the tips).

Order of operations

1What is 100 ÷ 10 ÷ 2?Use in quizUse

Answer:5(click to reveal)

Working: division goes left to right, so 100 ÷ 10 = 10, then 10 ÷ 2 = 5 (not 100 ÷ 5 = 20).

2What is 3 + 3 × 3 − 3 + 3?Use in quizUse

Answer:12(click to reveal)

Working: multiply first, 3 × 3 = 9, then 3 + 9 − 3 + 3 = 12.

3What is 10 − 4 ÷ 2 + 1 × 2?Use in quizUse

Answer:10(click to reveal)

Working: 4 ÷ 2 = 2 and 1 × 2 = 2, so 10 − 2 + 2 = 10.

4What is 2² + 2² × 2²?Use in quizUse

Answer:20(click to reveal)

Working: powers first, so it's 4 + 4 × 4, which is 4 + 16.

5What is 8 ÷ 2 × (2 + 2)?Use in quizUse

Answer:16(click to reveal)

Working: brackets first give 4, then left to right: 8 ÷ 2 = 4, and 4 × 4 = 16.

6Divide 30 by a half, then add 10. What do you get?Use in quizUse

Answer:70(click to reveal)

Working: dividing by a half is the same as doubling, so 30 ÷ ½ = 60, and 60 + 10 = 70.

Trick questions

7If you have 3 apples and take away 2, how many apples do you have?Use in quizUse

Answer:2(click to reveal)

Working: you took 2, so those are the ones you have.

8How many months of the year have 28 days?Use in quizUse

Answer:All 12(click to reveal)

Working: every month has at least 28 days.

9A farmer has 17 sheep. All but 9 run away. How many are left?Use in quizUse

Answer:9(click to reveal)

Working: "all but 9" means 9 stayed.

Brain teasers

10A bat and a ball cost £1.10 together. The bat costs £1 more than the ball. How much does the ball cost?Use in quizUse

Answer:5p (not 10p)(click to reveal)

Working: if the ball is 5p, the bat is £1.05, and together they make £1.10.

It's from the Cognitive Reflection Test, devised by Shane Frederick in 2005 to catch out gut answers, and so are the next two.

11If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?Use in quizUse

Answer:5 minutes(click to reveal)

Working: each machine makes one widget in 5 minutes, so 100 machines make 100 widgets in the same 5 minutes.

12A patch of lily pads doubles in size every day. It takes 48 days to cover the whole lake. How long does it take to cover half the lake?Use in quizUse

Answer:47 days(click to reveal)

Working: it doubles on the last day, so the day before, it covered half.

13A clock takes 5 seconds to strike six o'clock, timed from the first chime to the last. With the chimes evenly spaced, how long does it take to strike twelve?Use in quizUse

Answer:11 seconds(click to reveal)

Working: six chimes have five gaps, so each gap is 1 second. Twelve chimes have eleven gaps.

14A shop raises a price by 50%, then cuts the new price by 50%. Compared with the original, what's the overall change?Use in quizUse

Answer:25% lower(click to reveal)

Working: £100 goes up to £150, and half of £150 is £75.

15What is 0.1 + 0.2? (Be precise.)Use in quizUse

Answer:0.3(click to reveal)

Working: one tenth plus two tenths is three tenths. Give a bonus point for 0.30000000000000004, which is what many programming languages print because computers store decimals in binary (Python's documentation explains why).

16What is the next number in this sequence: 1, 11, 21, 1211, 111221, ?Use in quizUse

Answer:312211(click to reveal)

Working: each term describes the one before. 111221 is "three 1s, two 2s, one 1", which gives 31 22 11. It's called the look-and-say sequence.

The really hard ones

17What do you get if you add up every whole number from 1 to 100?Use in quizUse

Answer:5050(click to reveal)

Working: pair them off: 1 + 100, 2 + 99, 3 + 98 and so on gives 50 pairs of 101.

A popular story has a young Carl Friedrich Gauss spotting this at school, though it may well be legend.

18What's the smallest number of people you need in a room for there to be a better than 50% chance that two of them share a birthday?Use in quizUse

Answer:23(click to reveal)

Working: the chance that 23 people all have different birthdays (365/365 × 364/365 × ... × 343/365) is about 49.3%, so a shared one is about 50.7% (source).

19On a game show there are three doors: a car behind one, goats behind the others. You pick a door. The host, who knows where the car is, opens one of the other doors to show a goat, then offers you the chance to switch. Should you switch?Use in quizUse

Answer:Yes (switching wins two times in three)(click to reveal)

Working: your first pick is right only 1 time in 3. When it's wrong, the host has to leave the car behind the other closed door, so switching wins the other 2 times in 3.

When Marilyn vos Savant gave this answer to the Monty Hall problem in her magazine column in 1990, around 10,000 readers wrote in, nearly 1,000 of them with PhDs, mostly to say she was wrong.

20What is 0.999... (with the 9s recurring forever) equal to?Use in quizUse

Answer:1(click to reveal)

Working: ⅓ = 0.333..., and multiplying both sides by 3 gives 1 = 0.999... (source).

Tips for running a tricky maths round

Leave the viral sums out, or play them for laughs. The infamous "8 ÷ 2(2 + 2)" has two defensible answers, 16 and 1, depending on whether 2(2 + 2) counts as one unit. Academic writing often gives that implied multiplication priority, and even calculators disagree: a Casio and a Texas Instruments model give different answers to 6 ÷ 2(1 + 2) (source). That's why Q5 uses an explicit × sign. If you want the debate, ask the ambiguous version as a no-points bonus.

Ban phones and calculators, and set a time limit. Most of these are easy given five minutes and a pen; 30 seconds per question is what makes them tricky. When you reveal the answers, read out the working too. People don't mind losing a point once they see why, and it heads off most arguments.

Decide on near misses in advance. For Q10, accept 5p, 5 pence or £0.05. For Q14, "it goes down by 25%" and "it ends at 75% of the original" are the same answer. For Q19, "switch" is enough, no reasoning required. For Q8, don't accept "one".

Tiebreaker. Ask: "How many squares of any size are there on a chessboard?" The answer is 204 (64 one-by-one squares, 49 two-by-two, and so on down to the single eight-by-eight board). Expect plenty of 64s. Closest answer wins.

For more "wait, really?" moments, try our true or false quiz questions, and build any of these into a round with our pub quiz maker.