20 math brain teasers with answers, from quick sums to real head-scratchers

Twenty maths brain teasers in three flavours: quick sums with a shortcut, number patterns with a rule to find, and the classics that catch everyone once. Each answer is hidden until you click, and each comes with the working. No calculator, no algebra beyond a line or two.

Quick sums with a trick

What is 15% of 80, without a calculator?
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12
10% of 80 is 8. 5% is half of that, 4. Add them: 12. Break awkward percentages into 10% and 5% pieces.
I double a number and add 10. The result is 46. What was the number?
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18
Work backwards: 46 − 10 = 36, then 36 ÷ 2 = 18.
What is 49 × 51?
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2,499
The numbers straddle 50: (50 − 1)(50 + 1) = 2,500 − 1. The difference of squares saves you the long multiplication.
Which is bigger: two thirds of three quarters, or three quarters of two thirds?
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They are equal
Multiplication is commutative. Both are 6⁄12, which is a half.
Add the whole numbers from 1 to 100 in your head.
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5,050
Pair 1 with 100, 2 with 99, and so on. Fifty pairs, each summing to 101: 50 × 101 = 5,050.
What is 1,000 minus 1 minus 2 minus 3 … all the way down to minus 10?
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945
1 to 10 sums to 55. 1,000 − 55 = 945.
A shirt is reduced by 20%, then the sale price is increased by 20%. Is it back to the original price?
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No, it is 4% cheaper
£100 → £80 → £96. The second 20% is taken of a smaller number. Percentage changes do not cancel.

Number patterns

Fill the gap: 3, 6, 11, 18, 27, ?
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38
The gaps are 3, 5, 7, 9, so the next gap is 11. Equivalently the sequence is n² + 2.
What comes next: 1, 4, 9, 16, 25, 36, ?
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49
The square numbers. 7 × 7 = 49. The gaps are the odd numbers, which is a quick tell.
Continue the pattern: 100, 50, 25, 12.5, ?
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6.25
Halving each time. Decimals do not change the rule.
What is the missing number: 2, 6, 12, 20, 30, ?
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42
The gaps are 4, 6, 8, 10, so the next is 12. Each term is n × (n + 1): 6 × 7 = 42.
Which number does not belong: 4, 9, 16, 21, 25?
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21
The others are perfect squares. 21 is 3 × 7.
What comes next: 1, 2, 6, 24, 120, ?
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720
Multiply by 2, then 3, then 4, then 5, then 6. These are the factorials.
Find the rule: 5, 8, 14, 26, 50, ?
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98
Double and subtract 2 each time: 50 × 2 − 2 = 98. The gaps also double: 3, 6, 12, 24, 48.

The classics

A lily pad doubles in size every day and covers a pond in 30 days. On which day did it cover half the pond?
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Day 29
If it doubles daily, it was half the size the day before it was full. Not day 15.
Three friends split a £30 bill, £10 each. The waiter realises the bill was £25, returns £5, and the friends take £1 each and tip £2. Each paid £9, so £27, plus the £2 tip is £29. Where is the missing pound?
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There is no missing pound
The £27 the friends paid already includes the £2 tip (£25 bill + £2). Adding the tip again double counts it. The correct check is £27 = £25 + £2.
A brick weighs a kilogram plus half a brick. How much does a brick weigh?
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2 kg
If B = 1 + B⁄2, then B⁄2 = 1, so B = 2. Half a brick is a kilogram.
How many squares are there on a standard chessboard?
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204
Not 64. Count every size: 64 one-by-one, 49 two-by-two, 36, 25, 16, 9, 4 and 1 eight-by-eight. The sum of the squares from 1² to 8² is 204.
You have a 3-litre jug and a 5-litre jug and unlimited water. Measure exactly 4 litres.
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Fill 5, pour into 3, empty 3, pour the remaining 2 into 3, fill 5, top up 3
After pouring 5 into 3 you have 2 left in the big jug. Move it to the small jug, refill the big jug, and pour until the small jug is full: exactly 1 litre leaves, leaving 4.
In a knockout tennis tournament with 128 players, how many matches are played to find the winner?
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127
Every match eliminates exactly one player, and 127 players must be eliminated. No bracket arithmetic needed.

Keep going

If the quick sums slowed you down, the mental math guide teaches the left-to-right and round-and-adjust methods behind them, and the trainer drills them in sixty-second rounds. If the patterns were the fun part, paste any sequence into the sequence solver to see the rule named. For teasers across words and logic as well, see brain teasers for adults.

Common questions

How can I get better at math brain teasers?

Estimate before you calculate, look for a friendly number nearby, and check whether the puzzle is testing arithmetic or testing whether you read it carefully. Most of the teasers on this page reward a slow first read more than fast arithmetic.

Are these suitable for children?

The quick sums and patterns suit confident upper-primary and secondary students. The classics are more about reasoning than arithmetic and work well as family puzzles.

Where can I practise the arithmetic itself?

Brisk’s mental math trainer runs sixty-second timed rounds at three levels, and the daily challenge includes two quick mental maths questions every day.