40 puzzles6 categories · worked solutions

    Investment banking brainteasers

    The classic puzzles banks use to test how you think under pressure. Read each one, work it out, then flip the card to reveal the solution.

    Logic & lateral thinking

    Spot the trick or the structure.

    Solution

    Each man plays different opponents and does not play against each other; therefore, they can each have the same number of wins without any draws.

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    Solution

    You need to draw three socks to ensure a matching pair. With only two socks, you could end up with one blue and one black, but a third sock guarantees at least one matching pair.

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    Solution

    The task takes 8 days to reach 25% completion. This is because the completion rate doubles each day: 100% is achieved on day 10, 50% on day 9, leading to 25% on day 8.

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    Solution

    Turn switch 1 on for a few minutes, then turn it off. Turn switch 2 on and enter the room. The bulb that is lit is controlled by switch 2. The bulb that is off but warm to the touch is controlled by switch 1. The remaining cold, unlit bulb is controlled by switch 3.

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    Solution

    17 minutes. The Analyst and Associate cross together (2 min). The Analyst returns with the torch (1 min). The VP and MD cross together (10 min). The Associate returns with the torch (2 min). The Analyst and Associate cross again (2 min). Total: 2 + 1 + 10 + 2 + 2 = 17 minutes.

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    Solution

    Pick a fruit from the box labeled 'Mixed.' Since all labels are wrong, this box actually contains only one type of fruit. If you pick an apple, it is the Apples box. The box labeled 'Apples' cannot be Apples (wrong label), and since you already identified the Apples box, it must be Oranges. The remaining box labeled 'Oranges' is the Mixed box.

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    Solution

    Ask either guard: 'If I asked the other guard which door leads to freedom, what would he say?' Then choose the opposite door. The truthful guard will honestly report the liar's wrong answer (pointing to the death door), and the liar will lie about the truthful guard's correct answer (also pointing to the death door). Either way, both point to the wrong door, so pick the other one.

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    Solution

    Take any 10 coins and put them in a separate group, then flip all 10 over. If your group originally had x heads, flipping turns them into (10 - x) heads. The remaining pile also has (10 - x) heads. Both groups now match. For example, if you grabbed 3 heads and 7 tails, flipping gives 7 heads and 3 tails, while the other pile also has 7 heads.

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    Solution

    Pirate 1 keeps 98 coins, gives 1 coin each to Pirates 3 and 5, and nothing to Pirates 2 and 4. Working backward: with 2 pirates, Pirate 4 takes all 100. With 3 pirates, Pirate 3 gives 1 coin to Pirate 5 (who would get 0 otherwise). With 4, Pirate 2 gives 1 coin to Pirate 4. So Pirate 1 buys the cheapest votes: Pirates 3 and 5 each accept 1 coin since they would get 0 if Pirate 1 is eliminated.

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    Solution

    Yes, C's hat is black. If A saw two white hats on B and C, A would know their own is black (only 2 white hats exist). Since A did not know, B and C are not both wearing white. B knows this. If B saw a white hat on C, B could deduce their own must be black. Since B also did not know, C cannot be wearing white. Therefore, C is wearing black.

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    Probability

    Odds, expected value, and paradoxes.

    Solution

    Place one white marble in one jar and the rest of the marbles in the other jar. This way, you have a 50% chance of picking the jar with one white marble and a nearly 50% chance of picking a white marble from the other jar.

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    Solution

    The proportion is 1:1. Each birth is independent of the previous ones and has a 50% chance of being a boy or a girl.

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    Solution

    The probability of rolling a 6 is 1/6. Rolling it three times in a row is (1/6) * (1/6) * (1/6) = 1/216.

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    Solution

    Yes, you should always switch. Your initial pick had a 1/3 chance of being the offer. The remaining unopened envelope now has a 2/3 chance. Switching doubles your probability of getting the job offer. This is known as the Monty Hall Problem.

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    Solution

    Pull the trigger again. After surviving, you were in one of 4 safe chambers. Of those 4 positions, only 1 is followed by a loaded chamber, giving you a 1/4 (25%) chance of dying. Spinning the cylinder resets your odds to 2/6 (33%). Pulling again without spinning is safer.

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    Solution

    About 1%. Out of 10,000 people, 1 has the disease and tests positive (true positive). Of the remaining 9,999 healthy people, 1% (about 100) will falsely test positive. So out of roughly 101 positive results, only 1 truly has the disease: 1/101 is approximately 0.99%. This is called the false positive paradox and demonstrates Bayes' Theorem.

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    Solution

    Only 23 people. The probability that all n people have different birthdays is (365/365) x (364/365) x (363/365) x ... x ((365-n+1)/365). At 23 people, this drops below 50%, meaning a shared birthday becomes more likely than not. This counter-intuitive result is known as the Birthday Paradox.

    Flip back

    Solution

    2 flips. Let E be the expected number of flips. On each flip, there is a 1/2 chance of heads (done in 1 flip) and a 1/2 chance of tails (wasted flip, then start over). So E = (1/2)(1) + (1/2)(1 + E). Solving: E = 1/2 + 1/2 + E/2, which gives E/2 = 1, so E = 2.

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    Mental math

    Compute fast and show your work.

    Solution

    You buy 15 chocolates, eat them, and get 15 wrappers you exchange for 15 / 3 = 5 chocolates. You eat the 5 chocolates and exchange 3 wrappers for one more. You're left with 1 chocolate and 2 wrappers, you eat the chocolate and have enough wrappers to get one last chocolate. In total, you can eat 22 chocolates.

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    Solution

    7^4 = 2401. Start squaring 7: 7 * 7 = 49, then 7 * 49 = 7 * (50 - 1) = 350 - 7 = 343, finally 7 * 343 = 7 * (350 - 7) = 2450 - 49 = 2401.

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    Solution

    We know that the sum of all the numbers from 1 to n is: n * (n + 1) / 2, so we can get the sum of numbers from 1 to 50 (1275). To find the missing integer we have to sum the numbers from the excel sheet and substract the sum from 1275.

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    Solution

    5,050. Pair the numbers from opposite ends: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on. There are 50 such pairs, so the sum is 50 x 101 = 5,050. This technique is attributed to the mathematician Carl Friedrich Gauss, who reportedly discovered it as a child.

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    Solution

    6,272. Use the distributive property: 64 x 98 = 64 x (100 - 2) = 6,400 - 128 = 6,272. Breaking numbers into round components makes mental math much easier. This technique of rounding to the nearest hundred and subtracting is a common mental math strategy tested in finance interviews.

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    Speed, rates & motion

    Distance, time, and the hidden shortcut.

    Solution

    The snail reaches 27 feet at the end of the 27th day and climbs up the remaining 3 feet on the 28th day, so it takes 28 days.

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    Solution

    You can drive 25,000 miles: (5 wheels * 20,000) / (4 wheels), but only if you replace a wheel every 5,000 miles.

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    Solution

    It is impossible. To average 60 mph over 2 miles, the car would need to complete both laps in 2 minutes total. But the first lap at 30 mph already took exactly 2 minutes, leaving zero time for the second lap. No matter how fast you drive, you cannot make up the lost time.

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    Solution

    120 miles. The trains close the 150-mile gap at a combined speed of 150 mph, so they meet in exactly 1 hour. The bird flies continuously at 120 mph for that entire hour, covering 120 miles. The trick is to focus on total time rather than trying to sum the infinite back-and-forth trips.

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    Counting & numbers

    Combinatorics and number sense.

    Solution

    After removing one layer, the cube now measures 8 x 8 x 8, so 8^3 = 512 cubes remain.

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    Solution

    At 3:15, the minute hand is at the 3, but the hour hand is 1/4 of the way between the 3 and the 4. Each hour represents 30 degrees (360/12). Thus, 1/4 of 30 degrees is 7.5 degrees. The angle between the hands is 7.5 degrees.

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    Solution

    The next palindromic number after 72927 is 73037, so you would need to travel 110 miles. Driving further will increase the number in the 5th position, so you'll need to reach 3 in the 2nd position in order to get a new palindromic number. When you reach 73000, you'll need to drive 37 miles to get 73037.

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    Solution

    The total number of handshakes is given by the combination formula C(n, 2), where n is the number of guests. So, C(7, 2) = 7! / (5! * 2!), which can be simplified to (6 * 7) / 2 = 21 handshakes.

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    Solution

    11 times. The minute hand laps the hour hand once every 12/11 hours (approximately 65 minutes and 27 seconds). In 12 hours, the minute hand completes 12 revolutions while the hour hand completes 1, so the minute hand overtakes the hour hand 12 - 1 = 11 times, not 12, because the overlap at 12:00 at the start and after 12 hours is the same event.

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    Solution

    23 breaks. Each break increases the total number of pieces by exactly 1, regardless of how you choose to break. Starting with 1 piece and ending with 24 pieces always requires exactly 24 - 1 = 23 breaks. No clever breaking strategy can reduce this number.

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    Measuring & weighing

    Jugs, scales, and clever constraints.

    Solution

    Start both hourglasses. When the 4-minute hourglass runs out, flip it. The 7-minute hourglass has 3 minutes left at this point. When the 7-minute hourglass runs out, flip it, and leave the 4-minute hourglass running with its last minute of sand. When that minute runs out (8 minutes elapsed), the 7-minute hourglass has 1 minute of sand at the bottom: flip it, and when it empties 9 minutes have elapsed.

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    Solution

    Fill the 5-liter jug. Pour from the 5-liter jug into the 3-liter jug until the 3-liter jug is full, leaving 2 liters in the 5-liter jug. Empty the 3-liter jug. Pour the 2 liters from the 5-liter jug into the 3-liter jug. Fill the 5-liter jug again, and pour into the 3-liter jug until it's full. You'll have exactly 4 liters in the 5-liter jug.

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    Solution

    To distribute a 7 kg gold bar over seven days with only two cuts, first cut the bar into pieces of 1 kg, 2 kg, and 4 kg. Then, distribute the gold in this sequence: Day 1: give 1 kg, Day 2: exchange it for 2 kg (2 - 1 = 1), Day 3: give 1 kg, Day 4: exchange 4 kg for the 1 kg and 2 kg (4 - 2 -1 = 1), Day 5: give 1 kg, Day 6: exchange it for 2 kg (2 - 1 = 1), Day 7: give 1 kg.

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    Solution

    Select 1 coin from the first bag, 2 from the second, 3 from the third, and so on, up to 5 coins from the fifth bag. Weigh this combined selection. If all coins were genuine, the total weight would be 1 + 2 + 3 + 4 + 5 = 15 grams. However, each counterfeit coin adds an extra 0.1 grams. The excess weight indicates the number of counterfeit coins, and since you've taken a different number of coins from each bag, this will point to the specific bag containing the counterfeits.

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    Solution

    Light the first wire at both ends, which will burn out in 30 minutes due to the doubled burning rate. Simultaneously, light one end of the second wire. When the first wire is completely burnt, light the other end of the second wire. This remaining half will burn in 15 minutes, totaling 45 minutes.

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    Solution

    14 drops. Drop the first egg from floors 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, then 100 (decreasing intervals). If it breaks at any point, use the second egg to test floors one by one from the previous safe floor. The decreasing intervals ensure the worst case is always 14 drops. This works because 14 + 13 + 12 + ... + 1 = 105, which covers all 100 floors.

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    Solution

    Divide the 9 coins into three groups of 3. Weigh group 1 against group 2. If they balance, the counterfeit is in group 3. If one side is heavier, the counterfeit is in that group. Take the suspect group of 3 and weigh 1 coin against another. If they balance, the third coin is counterfeit. If not, the heavier one is the counterfeit. Solved in just 2 weighings.

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