Brainteaser and puzzle interview questions
Brainteasers test whether you can find structure fast: counting, number theory, clever weighings, recursion and sequences. Each problem below has a short solution that shows the trick, not just the number.
1A bat and ball cost $1.10 together. The bat costs $1.00 more than the ball. How many cents is the ball?
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Answer: 5
Ball = x, bat = x + 1. 2x + 1 = 1.10, so x = 0.05.
2Ten people each shake hands once with every other person. How many handshakes happen?
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Answer: 45
C(10,2) = 45.
3What is 1 + 2 + 3 + … + 100?
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Answer: 5,050
50 pairs that each sum to 101: 5050.
4How many distinct arrangements are there of the letters in LEVEL?
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Answer: 30
5! / (2! × 2!) = 30.
5How many diagonals does a 10-sided polygon have?
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Answer: 35
10 × 7 / 2 = 35.
6What is 15% of 15% of 2000?
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Answer: 45
0.15 × 0.15 = 0.0225, and 0.0225 × 2000 = 45.
7How many trailing zeros does 50! have?
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Answer: 12
A trailing zero needs a factor of 10 = 2 × 5. Factors of 2 are plentiful, so count the 5s: multiples of 5 give one each, multiples of 25 an extra one, and so on. ⌊50/5⌋ = 10, ⌊50/25⌋ = 2. Total: 12.
8What is the smaller angle, in degrees, between the hour and minute hands at 1:30?
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Answer: 135
Hour hand at 45°, minute hand at 180°. Smaller angle: 135°.
9How many distinct arrangements are there of the letters in BANANA?
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Answer: 60
Start with all 6! = 720 orderings, as if every letter were different. Swapping identical letters doesn't create a new word, so divide by 3! for the 3 As and 2! for the 2 Ns: 720 / 12 = 60.
10What is the sum of all integers from 1 to 50?
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Answer: 1,275
Pair the first and last terms: 1 + 50 = 51, and every pair moving inward has the same sum. That's 50 terms averaging 25.5, so the sum is 50 × 25.5 = 1,275.
11What is the sum of the first 5 odd numbers?
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Answer: 25
Try small cases: 1 = 1², 1 + 3 = 2², 1 + 3 + 5 = 3². Each new odd number turns a k × k square into a (k+1) × (k+1) square, so the first 5 odd numbers sum to 5² = 25.
12How many squares of any size are on an 2 × 2 board?
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Answer: 5
Count by size: a k × k square fits in (2 − k + 1)² positions. Adding sizes 1 to 2 gives 2² + 1² + … + 1² = 5.
136 people each shake hands once with every other person. How many handshakes happen?
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Answer: 15
Each handshake is one pair of people, so count pairs: 6 × 5 / 2 = 15. (Each person shakes 5 hands, and dividing by 2 stops each handshake being counted twice.)
14How many trailing zeros does 100! have?
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Answer: 24
⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24.
15What is the angle in degrees between the hour and minute hands at 3:15?
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Answer: 15/2
Minute hand at 90°. Hour hand at 90° + 15 × 0.5° = 97.5°. Difference 7.5°.
16100 closed lockers. Student k toggles every k-th locker, for k = 1..100. How many lockers are open at the end?
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Answer: 10
Only perfect squares have an odd number of divisors: 1, 4, …, 100 → 10 lockers.
17How many rectangles of any size are on an 8×8 chessboard?
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Answer: 1,296
C(9,2)² = 36² = 1296.
18Two trains 100 miles apart head toward each other at 50 mph each. A bird flies back and forth between them at 75 mph until they meet. How many miles does the bird fly?
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Answer: 75
The trains meet in 1 hour, so the bird flies 75 miles.
19How many times do the hour and minute hands overlap in 12 hours?
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Answer: 11
The minute hand gains a full lap 11 times every 12 hours.
20What is the last digit of 7^2026?
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Answer: 9
Last digits cycle 7, 9, 3, 1. 2026 leaves remainder 2 mod 4, so the digit is 9.
21What is the sum of the digits of all integers from 1 to 99?
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Answer: 900
Units: 10 × 45. Tens: 10 × 45. Total 900.
22A snail climbs 3 feet each day and slips 2 feet each night. How many days to escape a 10-foot well?
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Answer: 8
After 7 days it is at 7 feet; on day 8 it climbs to 10 and escapes.
23How many ways can you climb 10 stairs taking 1 or 2 steps at a time?
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Answer: 89
f(n) = f(n−1) + f(n−2): Fibonacci, f(10) = 89.
24What is the smallest positive integer divisible by every number from 1 to 10?
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Answer: 2,520
2³ × 3² × 5 × 7 = 2520.
25There are 100 doors and one prize. You pick a door; the host, who knows where the prize is, opens 98 empty doors you didn't pick. What is your probability of winning if you switch?
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Answer: 99/100 ≈ 0.99
Switching wins whenever the first pick was wrong: 99/100.
26Nine balls look identical, but one is heavier. With a balance scale, what is the minimum number of weighings that always finds it?
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Answer: 2
Weigh 3 against 3: the heavy ball is in the heavier group or the 3 left out. Then weigh 1 against 1 from that group. Two weighings, since 3² = 9.
27What is 1/(1·2) + 1/(2·3) + … + 1/(99·100)? (Decimal.)
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Answer: 99/100 ≈ 0.99
The sum telescopes to 1 − 1/100 = 0.99.
28What is the last digit of 2^99?
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Answer: 8
Only last digits matter. Powers of 2 end in 2, 4, 8, 6, then repeat (a cycle of 4). 99 leaves remainder 3 in that cycle, so 2^99 ends in 8.
29How many ways can you climb 5 stairs taking 1 or 2 steps at a time?
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Answer: 8
Look at the last move: it's either 1 stair (leaving 4 to climb before it) or 2 stairs (leaving 3). So f(n) = f(n − 1) + f(n − 2), with f(1) = 1 and f(2) = 2. Building up: 1, 2, 3, 5, 8. So f(5) = 8.
30What is the smallest positive integer divisible by every number from 1 to 4?
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Answer: 12
The answer needs the highest power of each prime that appears up to 4: 2^2 × 3 = 12.
31How many rectangles of any size are in a 2 × 3 grid of squares?
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Answer: 18
A rectangle is fixed by picking 2 of the 3 horizontal grid lines and 2 of the 4 vertical ones: C(3,2) × C(4,2) = 3 × 6 = 18.
32How many ways can you climb 4 stairs taking 1, 2 or 3 steps at a time?
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Answer: 7
Tribonacci recurrence: f(4) = 7.
- Look at the last move: it covers 1, 2 or 3 stairs. So f(n) = f(n−1) + f(n−2) + f(n−3).
- Base cases: f(0) = 1 (do nothing), f(1) = 1, f(2) = 2.
- Build up: f(3) = 4, f(4) = 7.
- So f(4) = 7.