Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
Hard: 29

Showing 1–4 of 4 · filtered from 100Clear filters

002A 10 x 10 x 10 cube is built from small unit cubes and the outside surface is painted green. How many small cubes have green paint on them, and how many have none?Mental maths and countingCoreTD SecuritiesNew York · 2024
Try it first
Quick instinct: how many small cubes are never painted?
Show the worked solution
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.
Each face of the big cube shows 8 corner cubes with three painted faces, 96 edge cubes with two and 384 face-centre cubes with one. They add to 488 painted cubes, and the 8 x 8 x 8 core of 512 cubes is never painted.
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.
The relationship
$\text{painted} = n^3 - (n-2)^3 = 1000 - 512 = 488$
n cubes along one edge, here 10
(n-2)^3 the hidden core after peeling one layer from each side
What it says in wordsThe painted cubes are everything minus the core you cannot see.
Where candidates lose it
The fast wrong answer is 600: six faces of 100. It counts every edge cube twice and every corner cube three times. Candidates who start from the surface usually end up correcting on the fly and lose the room.
Start from the inside. Say the core formula, give 512 and 488, then offer the corner, edge and face split as a check.
What the interviewer asks next
How many small cubes have exactly two painted faces?
What size cube has as many unpainted as painted small cubes, roughly?
Now paint only the top and bottom. How many cubes are painted?
Asked at TD Securities, Generalist, New York, 2024 (Wall Street Oasis): If I gave you a 10x10x10 cube and I painted the surface green, how many cubes are green?
005A stock falls 10% one day. How much does it need to rise the next day to get back to where it started?Mental maths and countingWarm upJefferiesLondon · 2026
Try it first
Pick the answer before you calculate.
Show the worked solution
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.
A fall from 100 to 90 is 10%, but the climb from 90 back to 100 is 11.1% because it starts from a smaller base. The gap widens with the size of the loss: 30% needs 42.9% to recover and 50% needs a full 100%.
The relationship
$g = \frac{L}{1-L} \qquad \frac{0.10}{0.90} = 11.1\%$
L the loss, as a decimal
g the gain needed to get back to the start
What it says in wordsThe gain needed to recover equals the loss divided by what is left after the loss.
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.
Where candidates lose it
Saying 10% is the whole trap, and it happens because the question is asked fast and sounds symmetric. The interviewer wants to see you notice the base changed.
If you say 11% rather than 11.1%, that is fine; say 10 over 90 so the method is audible.
What the interviewer asks next
A stock rises 50% and then falls 50%. Where does it end?
Two days of minus 10% and plus 10%: are you up or down, and by how much?
Why does this matter for how a fund reports its drawdowns?
Asked at Jefferies, Generalist (IBD spring week), London, 2026 (Wall Street Oasis): if a stock goes down 10% one day how much does it need to go up by the day after to get back
015Without a calculator, what is 9 to the power of 6?Mental maths and countingCoreNomuraTokyo · 2025
Try it first
Before computing anything, which size is right?
Show the worked solution
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.
Nine cubed is 729, and squaring 729 as 700 plus 29 gives a 490,000 block, two strips of 20,300 and an 841 corner, which add to 531,441.
The relationship
$9^6 = (9^3)^2 = 729^2 = (700+29)^2 = 490{,}000 + 2(700)(29) + 29^2 = 531{,}441$
9^3 729, nine cubed
(700 + 29)^2 the square split into a round part and a remainder
2(700)(29) the two strips, 20,300 each and 40,600 together
What it says in wordsSquare a number by splitting it into a round part and a remainder; the cross term counts twice.
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.
Where candidates lose it
The usual loss is multiplying 9 by itself six times in a row and dropping a digit somewhere in the five-digit middle steps. By the time you reach 59,049 times 9, the interviewer has watched you struggle for a minute.
The other loss is saying the number without a check. Give the answer, then the two-second test: it ends in 1 and its digits sum to 18.
What the interviewer asks next
What is 11 to the power of 4?
What are 99 squared and 999 squared?
Is 2 to the power of 20 bigger or smaller than a million, and by how much?
Asked at Nomura, Generalist, Tokyo, 2025 (Wall Street Oasis): I think one of the math questions was 9 to the power of 6
100Online numerical test style: three divisions had revenue of Rs 420, 260 and 120 crore last year and Rs 462, 299 and 150 crore this year. Which division grew fastest, and which contributed most to the growth in total revenue?Mental maths and countingWarm upBarclaysNew York · 2026
Try it first
Which division contributed most to total growth?
Show the worked solution
Division C grew fastest, at 25%, but Division A contributed most, adding Rs 42 crore of the Rs 111 crore increase. Growth rates are 42/420 = 10%, 39/260 = 15% and 30/120 = 25%. Rupee increases are 42, 39 and 30. A's slower rate on a much bigger base adds more. Total revenue grew 13.9%, and A supplied 37.8% of that growth.
Why are these two different questions?
A child's pocket money rising from 100 to 150 is 50% growth; a parent's salary rising from 1 lakh to 1.1 lakh is 10%. The household budget still cares far more about the salary. A growth rate measures change relative to each part's own size; contribution measures change in rupees, which is what adds up to the total. Online tests ask both in one question because candidates who rush answer the same division twice.
Adding a growth column and an increase column to the table shows Division C growing fastest at 25% while Division A adds the most, Rs 42 crore or 37.8% of the Rs 111 crore rise, and total revenue grows 13.9%.
How do you do it fast under a timer?
Compute the increases first, because they are subtractions: 42, 39, 30. Then compute rates only where needed, using easy fractions: 42 on 420 is a tenth, 30 on 120 is a quarter, 39 on 260 sits between them at 15%. Write the two added columns next to the table rather than holding the numbers in your head, because the next question in the set often reuses them.
The relationship
$g_{\text{total}} = \sum_i w_i\, g_i = \tfrac{420}{800}(10\%) + \tfrac{260}{800}(15\%) + \tfrac{120}{800}(25\%) = 13.9\%$
w_i each division's share of last year's revenue
g_i each division's growth rate
What it says in wordsTotal growth is the average of the divisions' growth rates weighted by their starting size.
That weighting is the third trap a test can set. The simple average of 10%, 15% and 25% is 16.7%, which is wrong because it gives the small division the same say as the large one. Total revenue went from 800 to 911, a rise of 13.9%. If an option shows 16.7%, it is there to catch exactly that shortcut.
Where candidates lose it
The trap is answering C to both parts, because the 25% is the most striking number on the page. The test separates fast readers from careful ones by asking for two different measures in one question.
The second loss is averaging growth rates to get total growth. Always weight by size, or simply add the totals and compute once.
What the interviewer asks next
If Division C keeps growing 25% and A keeps growing 10%, in how many years does C add more rupees than A?
What share of total revenue will each division have next year if growth rates repeat?
How would you present these numbers on one slide for a client?
Asked at Barclays, Investment Banking, New York, 2026 (Wall Street Oasis): The numerical section involved interpreting tables and charts quickly

n	cubes along one edge, here 10
(n-2)^3	the hidden core after peeling one layer from each side

L	the loss, as a decimal
g	the gain needed to get back to the start

9^3	729, nine cubed
(700 + 29)^2	the square split into a round part and a remainder
2(700)(29)	the two strips, 20,300 each and 40,600 together

w_i	each division's share of last year's revenue
g_i	each division's growth rate

Try it first

Why count the cubes you cannot see?

How do you check 488 the other way?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the recovery bigger than the fall?

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

Where candidates lose it

What the interviewer asks next

Try it first

How do you keep every step in your head?

How do you check it before saying it?

Where candidates lose it

What the interviewer asks next

Try it first

Why are these two different questions?

How do you do it fast under a timer?

Where candidates lose it

What the interviewer asks next