Investment Banking puzzles, solved step by step

488 small cubes are painted and 512 are not. The unpainted ones form a hidden core with one layer peeled off every side, so the core is 8 x 8 x 8, which is 512. The whole cube is 1,000, so 1,000 minus 512 leaves 488 with at least one painted face.

Why count the cubes you cannot see?

Picture a mango with its skin peeled. Measuring the skin piece by piece is fiddly; measuring the fruit left inside and subtracting from the whole is easy. The unpainted cubes form one clean block, so counting them and subtracting is faster and harder to get wrong than adding up the painted surface. Peeling one layer from both sides of each dimension takes 10 down to 8.

How do you check 488 the other way?

Count the surface directly, taking care not to double count. There are 8 corners, 12 edges each carrying 8 cubes once the corners are removed, and 6 faces each carrying an 8 x 8 centre. That gives 8 plus 96 plus 384, which is 488. Two methods agreeing is exactly what the interviewer wants to hear.

About 11.1%. A price of 100 falls 10% to 90. Getting back to 100 needs a rise of 10 on a base of 90, which is 10 divided by 90, or 11.1%. The general rule: to recover a loss of L, you need L divided by (1 minus L), which grows much faster than the loss itself.

Why is the recovery bigger than the fall?

A 10% discount on a 100 rupee shirt takes it to 90. If the shop then marks it up by 10%, the tag reads 99, not 100, because the markup is taken on 90. Percentages are always measured against a base, and after a fall the base is smaller, so the same rupee gain is a bigger percentage.

What do you add to sound like a banker rather than a calculator?

Give the number, then the pattern. Large drawdowns are hard to climb out of: a fund that loses 50% has to double just to break even. That one sentence shows you see why lenders and investors care about downside, which is the real reason the question is asked.

531,441. Rewrite 9 to the 6th as the square of 9 cubed. 9 cubed is 729. Then square 729 by splitting it into 700 plus 29: 700 squared is 490,000, twice 700 times 29 is 40,600, and 29 squared is 841. They add to 531,441. As a check, it should be a little over half a million, because 0.9 to the 6th is about 0.53.

How do you keep every step in your head?

Think of carrying a heavy box up six floors: you take it in two flights of three with a rest between, not six lifts in a row. Rewriting a power as the square of a smaller power cuts the work to one easy cube and one square you can split. 9 squared is 81, and 81 times 9 is 729, so 9 cubed is 729. Then 9 to the 6th is 729 squared, and squaring a number near a round one is the most practised move in mental maths.

How do you check it before saying it?

Three quick checks. An even power of 9 ends in 1, because 81 does, and 531,441 ends in 1. Any power of 9 is divisible by 9, so its digits must sum to a multiple of 9: 5 + 3 + 1 + 4 + 4 + 1 is 18. And the size is right: 9 is 10% below 10, and six 10% trims leave about 53% of a million. A digit check and a size check together catch nearly every slip in the cross term, which is where most mental errors happen.

A second route, if you prefer it: 9 to the 6th is also 81 cubed. 81 times 81 is 6,561, and 6,561 times 81 is 6,561 times 80 plus 6,561, which is 524,880 plus 6,561, giving 531,441 again. Saying one route and confirming with the other is worth more than speed.

115.6. Notice that 34% of 170 is the same as 17% of 340: halving one number and doubling the other leaves a product unchanged. So the sum is 17% of 340 twice, which is 34% of 340. That is 30% of 340, which is 102, plus 4% of 340, which is 13.6, giving 115.6.

What is the trick hiding in the numbers?

Two plots of land, one 34 metres by 17 and one 17 metres by 34, have the same area; turning a rectangle on its side does not change it. Percentages behave the same way, because x% of y is x times y divided by 100. The percentage and the base can trade places, so 34% of 170 equals 17% of 340: each is 17 x 340 / 100, which is 57.8. Once you see that, the question is one product, not two.

How do you do 34% of 340 in your head?

Split it into easy pieces: 30% of 340 is 102 and 4% of 340 is 13.6, so the total is 115.6. Or notice that 34% of 340 is 34 x 3.4, and 34 x 34 is 1,156, so the answer is 115.6. Spotting a structure first and calculating second is the habit the interviewer is checking, and it usually turns two awkward products into one easy one.

How do you check it before you say it?

Estimate first: 17% is a little more than a sixth, and a sixth of 340 is about 57, so each term is near 57 and the sum near 115. Then check each term directly: 10% of 340 is 34 and 7% is 23.8, so 17% is 57.8; 30% of 170 is 51 and 4% is 6.8, so 34% of 170 is 57.8. The two terms match, which confirms the swap.

50 teams. Choosing any 3 of 8 analysts gives 8 x 7 x 6 / 6 = 56 teams. The forbidden ones contain both analysts who refuse to work together, and the third seat can go to any of the other six, so there are 6 forbidden teams. 56 minus 6 leaves 50. Count everything, then subtract the cases the rule forbids.

Why count the teams you do not want?

Think of seating guests at a dinner when two of them have fallen out. Listing every acceptable table is slow; counting all possible tables and removing those that seat the two together is quick. When a rule forbids a small set of cases, count everything and subtract the forbidden ones, because the forbidden set is usually the easy one to count. Here a forbidden team is the two feuding analysts plus a third person, and there are only 6 choices for the third.

How do you get 56 without a formula sheet?

Fill the team one seat at a time: 8 choices, then 7, then 6, which is 336 ordered picks. Each team of three turns up in 3 x 2 x 1 = 6 different orders, so divide by 6 to get 56 distinct teams. That is the combinationA selection where order does not matter. The number of ways to choose k from n is written n choose k. count, 8 choose 3. Subtract the 6 forbidden teams and 50 remain.

How do you check 50 a second way?

Count the allowed teams directly, split by how many of the feuding pair they include. Teams with neither of the two are 6 choose 3, which is 20; teams with exactly one are 2 times 6 choose 2, which is 30; together 50. Two methods agreeing is the answer the interviewer remembers, and the second method is the one you will need when a follow-up adds a second rule.

66 handshakes. Each of the 12 bankers shakes hands with the other 11, which gives 12 x 11 = 132, but every handshake has two people in it and so has been counted twice. Half of 132 is 66. The same number falls out of adding 11 + 10 + 9 and so on down to 1, or from 12 choose 2.

Why do you halve the count?

Think of counting the phone calls in a group where everyone rang everyone else once. If each person reports the calls they took part in, every call shows up in two reports. Each handshake involves two people, so counting from each person's side counts every handshake exactly twice, and the true number is half. Twelve people times eleven others is 132 ends of handshakes, which is 66 handshakes.

How do you check 66 another way?

Let people arrive one at a time. The second person shakes 1 hand, the third 2, and so on to the twelfth, who shakes 11, and adding 1 through 11 gives 66. Pair the terms to add them quickly: 1 and 11, 2 and 10, and so on make five pairs of 12 plus a 6 in the middle, 66. Two routes to the same number is the check interviewers like to hear.

Where does this count show up on a desk?

Anywhere pairs matter. The number of pairs grows roughly with the square of the group: double the group to 24 and the handshakes rise to 276, more than four times as many. That is why a 12-stock portfolio has 66 pairwise correlationsMeasures of how closely two things move together, from minus 1 to plus 1. to estimate, and why a deal with many parties needs far more conversations than its headcount suggests.

204 squares. A square of side k can start in (9 minus k) columns and (9 minus k) rows, so there are 64 squares of size 1, 49 of size 2, 36 of size 3, and so on down to a single 8 x 8. The total is the sum of the first eight square numbers, 1 + 4 + 9 + ... + 64, which is 204.

How do you count one size without listing every square?

Picture sliding a 3 x 3 photo frame across the board. Its left edge can sit on column 1, 2, 3, 4, 5 or 6; on column 7 it would hang off the side. That is 6 positions across and, by the same logic, 6 down. Count where a square's top-left corner can sit, not the squares themselves: a k x k square has (9 minus k) choices in each direction, so (9 minus k) squared positions. For k = 3 that is 36, for k = 7 it is 4, and the whole board is the single 8 x 8.

How do you add 64 + 49 + ... + 1 quickly?

Add from the top and say the running total: 64 and 49 is 113, plus 36 is 149, plus 25 is 174, plus 16 is 190, then 9, 4 and 1 take it to 204. If you know the formula for a sum of squares, use it as a check rather than a starting point. The interviewer scores the moment you see that each size contributes a square number, not the formula you remember.

Why is 1,296 the wrong answer here?

1,296 is the number of rectangles of every shape. A rectangle is fixed by choosing two of the 9 vertical grid lines and two of the 9 horizontal ones, 36 ways each, and 36 x 36 is 1,296. Squares are the rectangles whose two sides are equal, which is why the count falls from 1,296 to 204. Offering that link unprompted shows you understand the counting rather than a memorised trick.

24 trailing zeros. Each zero at the end is a factor of 10, which is a 2 paired with a 5. Among the numbers 1 to 100 there are 97 factors of 2 but far fewer 5s, so the 5s decide it. The 20 multiples of 5 contribute one 5 each, and the four multiples of 25 contribute a second, giving 20 + 4 = 24. There is no multiple of 125 below 100, so there is no third layer.

Why is the answer about fives and not about tens?

Think of making pairs of shoes from a pile of lefts and rights. If you have 97 left shoes and 24 right shoes, you can make 24 pairs; the extra lefts do not help. A trailing zero is a pair of one 2 and one 5 multiplied together, and in 1 to 100 the 2s vastly outnumber the 5s, so the number of 5s sets the number of zeros. Counting multiples of 10 misses the zeros built from a 5 in one number and a 2 in another, such as 5 x 2 = 10 or 15 x 4 = 60.

Where do the extra four come from?

A number like 50 is 2 x 5 x 5: it carries two fives, not one. Every multiple of 25 hides a second five, and 25, 50, 75 and 100 are the four of them below 100, so the count is 20 from the multiples of 5 plus 4 from the multiples of 25. The next layer would be multiples of 125, which carry a third five, but 125 is beyond 100, so the count stops at 24. The pattern continues for larger factorials: divide by 5, then 25, then 125, and add the whole-number parts.

How do you check it without writing out 100 factorial?

Test the method on a factorial you can write. 10 factorial is 3,628,800, with two trailing zeros, and the rule gives 10 over 5 = 2, plus 10 over 25 = 0, which matches. Showing the rule on a small case you can verify is the fastest way to earn trust for the big case you cannot. Then say the general version: the number of trailing zeros of n factorial is the whole-number sum of n over 5, n over 25, n over 125 and so on, and for 1,000 factorial that is 200 + 40 + 8 + 1 = 249.

2,496 and 9,991. Both pairs sit the same distance either side of a round number, so use the difference of squares: (a minus b)(a plus b) is a squared minus b squared. 48 x 52 is 50 squared less 2 squared, 2,500 minus 4, 2,496. 97 x 103 is 100 squared less 3 squared, 10,000 minus 9, 9,991. Say the middle number and the gap out loud, then subtract.

Why is the product always a little less than the middle number squared?

Take a square courtyard 50 paces on each side, 2,500 square paces. Cut a strip 2 paces wide off one side and lay it along the bottom. The courtyard is now 48 by 52, but the strip you moved was only 48 long, so a 2 by 2 corner is left bare. Two numbers either side of a round one multiply to the round number squared minus the gap squared, because the strip you move never quite fills the corner. 48 x 52 is 2,500 less 4, which is 2,496; the bare corner is the 4.

How do you spot when the trick applies?

Add the two numbers and halve: if the result is round, the trick is on. 48 plus 52 is 100, half is 50, gap 2. 97 plus 103 is 200, half is 100, gap 3. Any pair that straddles a round number is a difference of squares: 73 x 67 is 70 squared less 3 squared, 4,900 minus 9, 4,891. If the pair does not straddle a round number, fall back on splitting one factor: 48 x 53 is 48 x 52 plus 48, 2,496 plus 48, 2,544. The reflex to look for the structure first is the skill the question tests.

Why does a banking interview bother with this?

Because the same reflex speeds up every number you say in a meeting. A 4% discount on a Rs 1,040 crore valuation, or a share price of Rs 97 times 103 crore shares, is a difference of squares in disguise, and getting it in two seconds without a calculator buys you credibility. Say your working out loud as you go: middle, gap, square, subtract. The interviewer is listening for the method as much as the answer, and a candidate who mumbles the right number silently earns less than one who narrates the route.

5,898 dollars. The fee is 1.5% of Rs 5,00,000, which is Rs 7,500, leaving Rs 4,92,500 to convert. Divided by 83.50 that is about 5,898.2. A quick route: 83.5 x 6,000 is 5,01,000, so Rs 5 lakh buys about 5,988 dollars before the fee, and taking 1.5% off 5,988 gives the same 5,898.

What is the question really testing?

These questions sit in the timed numerical section of an online assessment, and the arithmetic is easy. What they test is whether you apply each adjustment to the right base under time pressure. Think of a restaurant bill with a 10% service charge: it is 10% of the food, not of the food plus tax, and getting the base wrong gives a number that looks almost right. The test designers put the almost-right numbers in the options, so the base you apply the fee to decides whether you pick the correct one.

Does it matter whether you take the fee before or after converting?

Not for a percentage fee. Taking 1.5% off the rupees and then converting, or converting and then taking 1.5% off the dollars, multiplies by the same two numbers in a different order: 5,988 x 0.985 is also 5,898. Order matters only for a flat fee or a fee charged on a different base; a percentage fee on the same amount commutes with the conversion. Knowing this lets you pick whichever order makes the mental maths easier.

Where does 5,900 come from? Dividing by 1.015 answers a different question: how much could you convert if the 1.5% were charged on top of the converted amount. That base is smaller than Rs 5 lakh, so the fee comes out as about Rs 7,389 instead of Rs 7,500. On a test with options two dollars apart, that is a wrong answer.