QUANT BRAINTEASER INTERVIEW QUESTIONS

Nobody expects you to have seen every puzzle. What an interviewer notices is whether your first move is a method or a guess.

These are original representative questions, not real or leaked firm questions. Several are variations on well-known puzzle types. That is on purpose, since knowing the type is a real advantage.

A brainteaser in an interview is rarely about the answer alone. The interviewer wants to see what you do in the first thirty seconds, before you know the answer. So after each solution below, look at the first move, not just the result.

Start with the trap

1. A snail climbs 3 metres up a 10-metre well each day and slides back 2 metres each night. On which day does it get out?
   Show worked solution

   The quick answer, 10, is wrong. After n − 1 full days and nights the snail is at n − 1 metres, and on day n it reaches n + 2. It gets out once n + 2 ≥ 10, so on day 8. The first move is to check the final day on its own, because the snail doesn't slide back after it gets out.

2. Which is larger: 2¹⁰⁰ or 3⁶⁰?
   Show worked solution

   Write both with the same exponent. 2^100 = 32^20 and 3^60 = 27^20, so 2¹⁰⁰ is larger. You never need the actual numbers.

Try small cases

If a problem involves 50 or 100 of something, solve it for 3, 4, and 5 first. The pattern usually shows up by then.

3. Fifty bulbs start off. For k = 1 to 50, person k flips the switch on every bulb whose number is a multiple of k. How many bulbs end up on?
   Show worked solution

   Bulb n is flipped once for each divisor of n. Divisors come in pairs, except when n is a perfect square. So only square-numbered bulbs are flipped an odd number of times: 1, 4, 9, 16, 25, 36, 49. That makes 7 bulbs. Checking bulbs 1 through 10 by hand shows the pattern within a minute.

4. How many zeros are at the end of 50!?
   Show worked solution

   Each trailing zero needs a factor of 10, and there are more 2s than 5s, so count the factors of 5: ⌊50/5⌋ + ⌊50/25⌋ = 10 + 2 = 12. Checking 10! (two zeros) first confirms the method.

Work backwards from the goal

5. You have a 4-litre jug, a 9-litre jug, and a tap. Measure exactly 6 litres.
   Show worked solution

   Work out what the step before the goal must look like. 6 is 9 − 3, so you need the 4-litre jug to have exactly 3 litres of space, which means it holds 1 litre. To get 1 litre: fill the 9, pour off 4 twice (leaving 1), and move that 1 litre into the empty 4-litre jug. Then fill the 9 and top up the 4. The 9-litre jug now holds 6 litres.

6. Four people must cross a bridge at night with one torch. At most two cross at a time, and a pair moves at the slower person's pace. Their crossing times are 1, 3, 6, and 8 minutes. What is the fastest total?
   Show worked solution

   Compare the two standard strategies. Using the fastest person to walk everyone across takes 3 + 1 + 6 + 1 + 8 = 19. Sending the two slowest together takes 3 + 1 + 8 + 3 + 3 = 18: 1 and 3 cross, 1 returns, 6 and 8 cross, 3 returns, 1 and 3 cross. So the answer is 18 minutes. The point is to compare both strategies instead of assuming the first one is best.

Count the information

Weighing puzzles and guessing games are really about how many outcomes each step can separate.

7. Thirty coins look identical, but one is heavier. With a balance scale, how many weighings do you need to be sure of finding it?
   Show worked solution

   Each weighing has three outcomes: left side heavy, right side heavy, or balanced. So k weighings can separate at most 3^k coins. Since 3^3 = 27 < 30, three weighings are not enough and four are. A common mistake is to answer 3 because 30 "feels close" to 27.

8. I pick a whole number from 1 to 100. You ask yes-or-no questions. How many do you need to be sure of finding it?
   Show worked solution

   Each answer halves the possibilities at best, and 2^6 = 64 < 100 ≤ 128 = 2^7, so the answer is 7.

Set up a rate

9. What is the angle between the clock hands at 3:40?
   Show worked solution

   The minute hand moves 6° per minute, so it is at 240°. The hour hand moves 0.5° per minute, so it is at 90 + 20 = 110°. The difference is 130°.

10. After 12:00, when do the hands next line up?
    Show worked solution

    The minute hand gains 5.5° per minute on the hour hand, so it gains a full lap after 360 / 5.5 = 720/11 minutes. That is about 1:05:27. It follows that the hands line up 11 times in 12 hours, not 12, which makes a good follow-up to mention.

What to do in the room

Say which habit you are trying first, and why. If small cases don't show a pattern, say so and switch to another approach. An interviewer who hears "let me try n = 3 first" can see your method even if you don't finish.

Many brainteasers turn into probability questions halfway through. For those, see the probability question bank and the combinatorics guide.

Separate a lower bound from a construction

For the heavier-coin problem, four weighings are necessary because three outcomes per weighing can distinguish at most 27 candidates in three steps. They are sufficient too: weigh 10 coins against 10, retain the heavier group or the unweighed group, then split the remaining candidates into groups as evenly as possible. The largest groups shrink from 10 to 4 to 2 to 1 in the remaining three weighings.