Investment Banking puzzles, solved step by step

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

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046Fund A returns +30% and then -10%. Fund B returns +10% and then +10%. Starting with Rs 100 in each, which ends higher?Growth and compoundingWarm upMiddle market IBPrivate equity
Try it first
Which ends higher?
Show the worked solution
Fund B ends higher, at Rs 121 against Rs 117. Fund A goes to 130 and then loses 10% of 130, ending at 117. Fund B compounds 10% twice to 121. Both average 10% a year, but A's compound growth rate is only about 8.2% because its returns swing. Volatility drags on compound returns: roughly half the variance comes off the average each year.
Why does the same average give different endings?
Think of a salary that rises 30% one year and is cut 10% the next, against one that rises 10% twice. The cut lands on the higher salary, so it takes away more rupees than the same percentage would have earlier. Returns multiply rather than add, so a loss after a gain is taken from a bigger base, and an uneven path ends below a smooth one with the same average. Rs 100 grows to 130 and then gives back Rs 13; the steady fund never gives anything back.
Fund A rises to 130 and falls to 117 while Fund B climbs to 110 and then 121; both average 10% a year, but A compounds at only 8.2% because of its swing, so B ends Rs 4 higher.
How big is the drag, and can you estimate it in your head?
The growth rate that matters is the geometric averageThe constant yearly return that would turn the starting amount into the ending amount over the same period.. Fund A's is the square root of 1.17 minus 1, about 8.2%, against B's 10%, even though both arithmetic averages are 10%. A quick estimate: subtract half the variance. A's returns sit 20 points either side of 10%, so half of 0.2 squared is 2 points, and 10% minus 2% is about 8%, close to the exact 8.2%.
The relationship
$g_A = \sqrt{1.30 \times 0.90} - 1 \approx 8.2\% \qquad g_A \approx \bar r - \tfrac{\sigma^2}{2} = 10\% - 2\% = 8\%$
r-bar the arithmetic average return, 10%
sigma how far returns swing around the average, 20 points for Fund A
What it says in wordsCompound growth is roughly the average return minus half the variance.
Why would an interviewer care?
Because fund reports often quote average returns, and investors live on compound ones. Two funds with the same average return can leave an investor with very different money, and the more volatile one leaves less. The same arithmetic explains why a fund that rises 50% and then falls 50% is down 25%, and why leverage that doubles volatility can lower long-run growth even while it raises the average.
Where candidates lose it
The trap is answering that they end level, because both funds average 10% a year. Averaging percentages assumes they add, and returns multiply.
The second loss is getting 117 and 121 without saying why. Name volatility drag and give the half-the-variance estimate; that turns a calculation into an insight.
What the interviewer asks next
A fund rises 50% and then falls 50%. Where does it end?
What steady yearly return matches Fund A over the two years?
Fund C returns +40% and then -20%. How does it compare with A and B?
047A deal is agreed at a fixed exchange ratio of 0.5 acquirer shares for each target share, when the acquirer trades at Rs 200. Before closing, the acquirer's shares fall 10%. What do target holders now receive, and how would a fixed-price structure have differed?Deal mathsHardElite boutique IBPrivate equity
Try it first
Under the fixed ratio, what is a target share now worth?
Show the worked solution
Under the fixed ratio, target holders receive Rs 90 of value per share instead of Rs 100. They still get 0.5 acquirer shares, but each is now worth Rs 180. Under a fixed price they would still receive Rs 100, and the acquirer would issue 100 over 180, about 0.556 shares per target share, 11.1% more than planned. A fixed ratio shares the price risk with the target; a fixed price keeps it with the acquirer and its shareholders.
What does each structure actually fix?
Think of agreeing to swap your car for 50 grams of gold, rather than for Rs 5 lakh paid in gold. If the gold price falls before the swap, the first deal still hands you 50 grams, now worth less; the second hands you more grams so that you still get Rs 5 lakh. A fixed exchange ratioThe number of acquirer shares paid for each target share in a stock deal. fixes the number of shares the target receives; a fixed price fixes their rupee value, and the share count moves instead. Whichever number floats carries the risk of the acquirer's share price between signing and closing.
After the acquirer falls from Rs 200 to Rs 180, a fixed ratio still pays 0.5 shares, now worth Rs 90 per target share, while a fixed price still pays Rs 100 by issuing 0.556 shares per target share, 11.1% more than planned.
How do the two structures play out across the whole deal?
Assume the target has 10 crore shares and the acquirer 50 crore. Fixed ratio: target holders receive Rs 900 crore of value instead of Rs 1,000 crore, and own 5 crore of 55 crore shares, 9.1% of the combined company. Fixed price: they still receive Rs 1,000 crore, so the acquirer issues 5.56 crore shares, 11.1% more than planned, and the target's holders own 10.0%. The acquirer's existing shareholders give up more of the company to pay the same rupees.
Per target share unless stated Fixed ratio Fixed price
Acquirer shares received 0.500 0.556
Value at Rs 180 Rs 90 Rs 100
New acquirer shares, crore 5.00 5.56
Target holders' share of the combined company 9.1% 10.0%
With 10 crore target shares and 50 crore acquirer shares, the fixed ratio leaves target holders with Rs 90 a share and 9.1% of the combined company, while the fixed price gives them Rs 100 a share and 10.0%.
Which side wants which, and how do bankers split the difference?
A target worried that the acquirer's shares will fall wants a fixed price; an acquirer worried about issuing too many shares wants a fixed ratio. A collar is the usual compromise: in one common form the ratio is fixed while the acquirer's price stays inside a band and the value is fixed outside it, so each side carries the risk only up to a point. Interviewers ask this to see whether you think about who carries risk between signing and closing, not just the headline value.
Where candidates lose it
Candidates say target holders still get Rs 100, because that was the headline price. A fixed ratio never promised rupees, only shares, and the shares are now worth less.
The second miss comes under the fixed price: saying nothing changes. The acquirer must issue 11.1% more shares, so its own shareholders absorb the fall through extra dilution.
What the interviewer asks next
The acquirer's shares rise 10% instead. Which structure would the target have preferred?
Design a collar that protects the target against falls of more than 10%.
Why might an acquirer with a volatile share price prefer to pay cash?
048You have 1,000 bottles of wine and exactly one is poisoned. A single sip makes a taster ill after exactly 24 hours. With 10 testers and one day, how do you find the poisoned bottle?Logic and brainteasersHardConsulting style brainteasersSales and trading
Try it first
How can 10 testers cover 1,000 bottles in one round?
Show the worked solution
Number the bottles 1 to 1,000, write each number in 10-digit binary, and have tester k sip from every bottle whose kth digit is a 1. After 24 hours, write a 1 for each tester who is ill and a 0 for each who is not. That 10-digit binary number is the poisoned bottle. It works because 10 yes-or-no results can tell apart 2 to the 10th, 1,024 cases, which is more than 1,000.
Why does giving each tester a separate batch fail?
Picture 10 friends each tasting their own 100 bottles. One friend falls ill and you know the poison is among their 100, but not which one, and the day is over. When testers taste separate batches, each result answers one question about one batch, so 10 testers can single out at most 10 groups, not 1,000 bottles. The fix is to let testers overlap, so that every bottle is sipped by its own unique combination of testers.
Writing each bottle number in 10-digit binary assigns one digit to each tester, so bottle 357, which is 0101100101, is sipped by testers 2, 4, 5, 8 and 10, and if exactly those five fall ill their place values add back to 357.
How does the binary code name the bottle?
Give each tester a place value: tester 1 is worth 512, tester 2 is worth 256, and so on down to tester 10, worth 1. Bottle 357 equals 256 + 64 + 32 + 4 + 1, so testers 2, 4, 5, 8 and 10 sip from it. If exactly those five fall ill, adding their place values gives back 357, and no other bottle has the same set of tasters. Every bottle number is a different set of testers, so the pattern of illness points to one bottle only.
The relationship
$2^{10} = 1{,}024 \ge 1{,}000 \qquad 357 = 256 + 64 + 32 + 4 + 1 = 0101100101_2$
2^10 the number of different ill-or-well patterns 10 testers can show
0101100101 bottle 357 in binary, one digit per tester
What it says in wordsTen yes-or-no results give enough patterns to label every bottle uniquely.
What is the general lesson?
Each tester is one yes-or-no question asked of every bottle at once. Ten yes-or-no answers can tell apart 1,024 possibilities, so the right design makes each answer carry one binary digit of the bottle's number. The same idea finds one faulty item among many with very few tests, and it is why 2,000 bottles need only one more tester. With two days, each tester has three outcomes, ill on day one, ill on day two, or never, and 7 testers would be enough.
Where candidates lose it
Most candidates split the bottles into 10 batches of 100, find the right batch, and run out of time. Separate batches waste the testers, because each one answers only a single question about a single batch.
The second loss is saying binary without showing the decoding. Work one bottle through, such as 357, so the interviewer sees the pattern of illness turn back into a number.
What the interviewer asks next
How many testers would you need for 2,000 bottles?
With two poisoned bottles, why does this scheme break down?
You have two days instead of one. How few testers can you manage with?
050A screening test flags accounting fraud with 95% accuracy both ways: it flags 95% of companies that commit fraud and clears 95% of companies that do not. In the sample, 2% of companies commit fraud. A company is flagged. What is the probability it is actually committing fraud?ProbabilityHardBulge bracket IBSales and trading
Try it first
Instinct first: how likely is a flagged company to be a fraud?
Show the worked solution
About 28%. Take 1,000 companies. Twenty commit fraud and the test flags 19 of them. The other 980 are clean, and the test wrongly flags 5% of them, which is 49. So 68 companies are flagged and only 19 are frauds: 19 over 68 is 27.9%. The test is accurate, but fraud is rare, so a small error rate on the large clean group produces more false flags than true ones.
Why is the answer not 95%?
Think of a smoke alarm that sounds for 95% of real fires and stays quiet 95% of the time there is no fire. In a building where fires are rare, most alarms you hear are burnt toast, because the quiet days vastly outnumber the fires and 5% of them still ring. A test's accuracy tells you how it behaves when the truth is known; the question asks the reverse, how likely the truth is given the test, and the base rateHow common the thing is before any test is run. Here, 2% of companies commit fraud. sits between the two. With 2% frauds, a 5% error on the 98% of clean companies creates 49 false flags for every 19 true ones.
Of 1,000 companies, the test flags 19 of the 20 frauds and 49 of the 980 clean companies, so 68 are flagged and only 19 of them are frauds, which makes a flag 28% likely to be right rather than 95%.
How do you work it with whole numbers?
Start with 1,000 companies and walk them down the tree. Frauds: 2% of 1,000 is 20, and 95% of those, 19, are flagged. Clean: 980, and 5% of those, 49, are wrongly flagged. 19 true flags and 49 false flags make 68 flagged companies, so the chance that a flagged company is a fraud is 19 over 68, about 28%. Working in counts rather than percentages is faster in the room and harder to get wrong, and it shows the interviewer where every number came from.
The relationship
$P(\text{fraud} \mid \text{flag}) = \frac{0.02 \times 0.95}{0.02 \times 0.95 + 0.98 \times 0.05} = \frac{0.019}{0.068} \approx 0.28$
0.02 x 0.95 the share of all companies that are frauds and get flagged
0.98 x 0.05 the share that are clean and get flagged anyway
What it says in wordsDivide the true flags by all the flags, true and false together.
What moves the number, and what does that mean for a screen?
Two things. The base rate: if 10% of the sample committed fraud, the same test would make a flag 68% reliable. The false-flag rate: cutting it from 5% to 1% lifts the answer to 66% even at a 2% base rate. A screen for a rare event is a filter that earns a closer look, not a verdict, and the second look does the real work: run an independent test of the same accuracy on the 68 flagged companies and a double flag is 88% likely to be a fraud. That is how diligence red flags, transaction monitoring and credit early-warning lists are meant to be read. The working assumes the two tests err independently; if they look at the same ratios, the second adds less than the arithmetic suggests.
Where candidates lose it
Candidates answer 95%, reading the test's accuracy as the answer. The question asks the reverse conditional, and when the event is rare the two numbers can sit far apart.
The second loss is reaching for the formula in symbols and getting lost in it. Say 1,000 companies, 20 frauds, 19 caught, 49 false flags, 19 over 68; the counts carry you through and the interviewer can follow every step.
What the interviewer asks next
The company is also flagged by a second, independent test with the same accuracy. Now what is the probability?
How rare does fraud have to be before a flag is more likely wrong than right?
The test clears a company instead. What is the chance it is actually clean?
051Estimate the number of two-wheelers sold in India in a year.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Pick a ballpark before you build it.
Show the worked solution
About 2 crore a year. India has about 140 crore people in roughly 30 crore households. If about half of households own a two-wheeler, 15 crore are in use. A vehicle lasts about 10 years, so 1.5 crore are replaced each year, and a fleet growing 3% adds about 45 lakh more, close to 2 crore. A top-down check, 18 crore vehicles on the road over a 9-year life, lands at 2.0 crore too. The published industry figure is the thing to confirm, not to recall.
Where do you start, and what do you refuse to guess?
Think of guessing how many birthday cakes a town buys in a year. Nobody knows the number, but everybody can picture the pieces: people, birthdays per person, the share with a cake. Market sizing works from a number you can picture to the number you cannot, and the only figure taken on trust is the population. About 140 crore people at 4.5 per household is 31 crore households, call it 30. The one real judgement is the ownership rate, the penetrationThe share of possible buyers who already own the product. Half of households here.: higher in towns, lower in villages, so about half overall.
Working up from 30 crore households, half owning a two-wheeler that lasts 10 years, gives 1.95 crore sales a year once 3% fleet growth is added, and working down from 18 crore vehicles on the road over a 9-year life gives 2.0 crore, so the two routes agree within 4%.
How does the bottom-up build go?
Thirty crore households at 50% is 15 crore two-wheelers in use. Sales are what it takes to keep that fleet going plus what it takes to grow it. Yearly sales of a durable are the fleet divided by its life, plus the fleet's growth, so the life assumption matters as much as the ownership rate. At a 10-year life, 1.5 crore are replaced each year; a fleet growing 3% a year needs another 45 lakh; together about 1.95 crore. Say each assumption as you use it, so the interviewer can push on the one they disagree with.
The relationship
$S = \frac{H \times p}{L} + H \times p \times g = \frac{30 \times 0.5}{10} + 15 \times 0.03 \approx 1.95\text{ crore}$
H households, about 30 crore
p the share of households with a two-wheeler, 50%
L the vehicle's life, 10 years
g growth in the fleet, 3% a year
What it says in wordsSales replace the vehicles that wear out and add the ones that a growing fleet needs.
Why build a second route, and what if it disagrees?
Come at it from the fleet instead. About 13 two-wheelers per 100 people is 18 crore on the road, and dividing by a 9-year average life gives 2.0 crore a year, within 4% of the first route. A demand estimate is credible only once an independent route lands near it; agreement does not prove either number, but disagreement tells you which assumption to revisit. Stress the two judgements: 40% ownership and a 12-year life give 1.4 crore, 60% and 8 years give 2.8 crore. So the honest answer is a range of about 1.5 to 3 crore with 2 crore most likely, and the published sales figure from the industry body is the check to look up afterwards.
Where candidates lose it
Candidates reach for a remembered figure, right or wrong, and have nothing to say when asked how they got it. The interviewer is marking the structure and the stated assumptions, not the number.
The second loss is forgetting replacement. A fleet of 15 crore does not mean 15 crore sales; sales are the fleet divided by its life plus growth, which is why the life assumption deserves its own sentence.
What the interviewer asks next
How would the estimate change if you were asked for cars instead?
What share of those sales would you expect to be first-time buyers rather than replacements?
How would you estimate the number of two-wheeler dealerships in India?
053You roll a fair die and are paid its face value in lakh rupees. After seeing the first roll you may roll once more and take the second result instead, whatever it turns out to be. What is the game worth, and what is your stopping rule?Expected value and gamesCoreBulge bracket IBConsulting style brainteasers
Try it first
On which first rolls do you roll again?
Show the worked solution
The game is worth Rs 4.25 lakh, and you roll again on a 1, 2 or 3. A single roll is worth 3.5 lakh on average, so the chance to roll again is worth exactly that. Keep any first roll above 3.5, which means 4, 5 or 6. Half the time you keep, averaging 5 lakh; half the time you roll again for 3.5. Half of 5 plus half of 3.5 is 4.25 lakh. The option to re-roll adds 0.75 lakh over a single roll.
Why is the rule to compare against 3.5?
Think of boarding a train with one visible seat by the door and the rest of the carriage out of sight. You walk on only if the seat you can see is worse than the average seat down the carriage, because once you walk you take whatever is there. Work backwards: the last decision has no choices left, so its value is the plain average, 3.5 lakh, and any earlier roll is kept only if it beats that. A 4, 5 or 6 beats 3.5; a 1, 2 or 3 does not; no face equals it, so the rule has no ties. The number you are comparing with is the expected valueThe average outcome if the same gamble were repeated many times: each result weighted by its probability. of rolling again.
A first roll of 1, 2 or 3 falls below the 3.5 that a second roll is worth, so you roll again, while a 4, 5 or 6 is kept, and the kept faces contribute 2.5 and the re-rolls 1.75 for a game worth 4.25 lakh.
How does the 4.25 come together?
Three faces out of six are kept and contribute (4 + 5 + 6) over 6, which is 2.5 lakh; three are re-rolled and contribute three sixths of 3.5, which is 1.75; the total is 4.25 lakh. Check the rule by moving it. Keep the 3 as well: (3 + 4 + 5 + 6) over 6 plus two sixths of 3.5 is 4.17. Re-roll the 4 too: (5 + 6) over 6 plus four sixths of 3.5 is 4.17. Both sit below 4.25, about Rs 8,333 a game, so 3.5 is the only bar that gives the full value.
The relationship
$V = \frac{4 + 5 + 6}{6} + \frac{3}{6} \times 3.5 = 2.5 + 1.75 = 4.25$
(4 + 5 + 6)/6 the kept faces, each one in six
3/6 the chance the first roll is a 1, 2 or 3 and you roll again
3.5 what the second roll is worth on average
What it says in wordsAdd what you keep, weighted by its chance, to what you get from rolling again, weighted by its chance.
What happens with a third roll, and why do interviewers ask?
Allow two re-rolls. The game from the second roll onward is now worth 4.25, so on the first roll you keep only a 5 or 6: (5 + 6) over 6 plus four sixths of 4.25 is 4.67 lakh. Every extra chance to try again raises both the value of the game and the bar for stopping, and that is the whole logic of an option: the right to swap a known result for an uncertain one is worth paying for only when the known result is poor. The same reasoning prices walking away from a signed deal or waiting for a better bid. The limit: this assumes you care only about the average. Someone who badly needs 3 lakh might keep a 3, and that is a preference, not an error.
Where candidates lose it
The common miss is a rule based on feel, such as re-rolling anything below 5 because 5 and 6 feel like wins. That throws away a 4 that beats the 3.5 you get by rolling again, and costs about Rs 8,333 of value per game.
The second loss is valuing the game at 3.5 lakh because it is still just a die. The option to re-roll is worth 0.75 lakh here, and pricing an option at zero is the error the interviewer is listening for.
What the interviewer asks next
You may re-roll twice. What is the game worth and what is the rule on the first roll?
The re-roll costs Rs 50,000. Do you still take it, and when?
How does this connect to the decision to walk away from a signed deal?

Per target share unless stated	Fixed ratio	Fixed price
Acquirer shares received	0.500	0.556
Value at Rs 180	Rs 90	Rs 100
New acquirer shares, crore	5.00	5.56
Target holders' share of the combined company	9.1%	10.0%

054On the last day of the financial year, a company buys a Rs 100 crore machine on 60-day credit from the supplier. What changes on each of the three statements at year end?Accounting riddlesWarm upBulge bracket IBMiddle market IB

Try it first

Which statement moves at year end?

Show the worked solution

Only the balance sheet changes. Property, plant and equipment rises by Rs 100 crore and accounts payable rises by Rs 100 crore, so both sides grow by the same amount. No cash has moved, so the cash flow statement is untouched, and no time has passed for depreciation, so the income statement is untouched too. Cash and capex appear in 60 days, when the supplier is paid.

Why does a purchase on credit touch neither cash nor profit?

Think of buying a refrigerator on a shop's 60-day credit. The fridge is in your kitchen today and you owe the shop, but your bank balance has not moved and nothing has come out of this month's budget. A credit purchase adds an asset and a debt of the same size, so the balance sheet grows on both sides and nothing else moves. The machine will be used for years, so its cost is not an expense on the day it arrives. It reaches the income statement slowly, through depreciation, over the years it is used.

At year end the Rs 100 crore machine raises property, plant and equipment by 100 and accounts payable by 100, while the income statement and the cash flow statement show nothing. Cash and capex move only on day 60, when the supplier is paid.

What happens over the next 60 days and beyond?

Day 60 is when the cash flow statement wakes up. The company pays the supplier: cash falls by 100 and payables fall by 100. The Rs 100 crore appears as capital expenditure in investing cash flow in the year it is paid, not the year the machine arrived. The notes to the accounts usually flag the year-end purchase as a non-cash investing item, so a reader is not surprised. From the following year, depreciation starts: on a 10-year straight line, Rs 10 crore a year comes off pre-tax profit, is added back in operating cash flow, and lowers the machine's book value.

Moment	Balance sheet	Income statement	Cash flow statement
Year end, machine arrives	PP&E +100, payables +100	No change	No change
Day 60, supplier paid	Cash -100, payables -100	No change	Investing outflow -100
Each later year, 10-year life	PP&E -10	Depreciation -10 before tax	Depreciation added back

Rs crore. The same machine touches the balance sheet on day one, the cash flow statement on day 60 and the income statement only from the following year, through depreciation of Rs 10 crore a year.

Close by saying the balance check out loud. Assets up 100, liabilities up 100: the sheet balances, and that one sentence tells the interviewer you walk the statements in a fixed order rather than guessing. Interviewers use small timing questions like this one to see whether you separate when something is owned, when it is paid for and when it is expensed.

Where candidates lose it

The common slip is putting Rs 100 crore of capex on the cash flow statement at year end, because buying a machine feels like capex. The cash flow statement records cash paid, and the company has paid nothing yet.

The second slip is expensing the machine, or charging a full year of depreciation on day one. Say the timing out loud: asset and debt today, cash in 60 days, depreciation from next year.

What the interviewer asks next

Now the company pays cash on day one instead. Walk me through the three statements.
At the end of next year, with a 10-year life and a 25% tax rate, what has changed on each statement?
The machine turns out to be faulty and is returned before the invoice is paid. What reverses?

055How many squares of any size are there on an 8 x 8 chessboard?Mental maths and countingHardBulge bracket IBMiddle market IB
Try it first
Before you count: how many squares are on the board?
Show the worked solution
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.
A 3 x 3 square can start in 6 columns and 6 rows, 36 positions, and the same rule gives 64, 49, 36, 25, 16, 9, 4 and 1 squares for sizes 1 to 8, adding to 204 squares on the board.
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.
The relationship
$\sum_{k=1}^{8}(9-k)^2 = \sum_{m=1}^{8} m^2 = \frac{8\cdot 9\cdot 17}{6} = 204$
k the side of the square, from 1 to 8
9-k the starting positions in one direction for that size
m^2 the same sum written from the 8 x 8 upwards
What it says in wordsCount the positions for each size, then add the first eight square numbers.
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.
Where candidates lose it
The quick answers are 64 and 65. Both stop at the obvious squares and miss that a 2 x 2 or a 5 x 5 square can sit in many places. The question is rated hard precisely because the first answer feels complete.
The other loss is listing positions by hand for each size and running out of time. State the rule, (9 minus k) squared, once, then add the eight numbers out loud.
What the interviewer asks next
How many rectangles of any size are on the board?
What is the general formula for an n x n board?
How many squares are on a 10 x 10 board?
056A company is funded 60% by equity at a 14% cost and 40% by debt at 10% before tax, and the tax rate is 25%. What is its weighted average cost of capital?DCF and cost of capitalWarm upBulge bracket IBMiddle market IB
Try it first
Pick the WACC.
Show the worked solution
The WACC is 11.4%. Equity is 60% of the funding at 14%, contributing 8.4 points. Debt is 40% at 10% before tax, but interest is tax-deductible, so its after-tax cost is 10% x (1 minus 25%) = 7.5%, contributing 3.0 points. Add them: 8.4 + 3.0 = 11.4%. Leaving out the tax shield gives 12.4%.
Why is it a weighted average and not a plain one?
A family buying a flat with 60% from savings and 40% from a home loan pays a blended cost that leans towards the bigger source. Each source of money counts in proportion to how much of the funding it provides. Here equity provides 60 rupees in every 100, so its 14% carries more weight than debt's rate. A plain average of 14% and 10% would be 12.0%, which pretends the two sources are the same size.
Equity is 60% of funding at 14% and contributes 8.4 points; debt is 40% at 7.5% after tax and contributes 3.0 points, so the WACC is 11.4%. Using the 10% pre-tax rate adds a wrong extra point and gives 12.4%.
Why does debt go in after tax?
Interest is deducted before profit is taxed, so every 10 rupees of interest cuts the tax bill by 2.50 rupees at a 25% rate. The tax saved pays a quarter of the interest, so the company's true cost of debt is 7.5%, not 10%. That saving is the tax shieldThe tax a company avoids because interest is deducted from profit before tax is worked out.. Equity has no such shield, because dividends are paid out of profit after tax, which is why the adjustment sits on the debt term only.
The relationship
$\text{WACC} = \tfrac{E}{V}\,r_e + \tfrac{D}{V}\,r_d(1-t) = 0.6(14\%) + 0.4(10\%)(0.75) = 11.4\%$
E/V equity's share of total funding, 60%
D/V debt's share of total funding, 40%
r_e cost of equity, 14%
r_d pre-tax cost of debt, 10%
t tax rate, 25%
What it says in wordsWeight each source's cost by its share of the funding, and cut the cost of debt by the tax it saves.
One more sentence wins the point. The weights should be market values, not the book values on the balance sheet, because investors demand a return on what their stake is worth today. Then say what the number is for: 11.4% is the rate at which you would discount the company's unlevered free cash flows in a DCF.
Where candidates lose it
The usual miss is forgetting the tax shield and answering 12.4%. It is the most common one-point error in a first-round cost of capital question, and interviewers ask it because it is so easy to skip.
The second is applying the tax adjustment to equity as well, or to the whole WACC. The shield belongs to interest alone, so say which term it sits on.
What the interviewer asks next
The company moves to 60% debt at the same rates. What happens to WACC, and why would the rates not stay the same?
Why is the cost of equity higher than the cost of debt?
If the company makes losses and pays no tax, what is its WACC?
057Logical test style: what number comes next in 2, 6, 12, 20, 30, and why?Logic and brainteasersWarm upConsulting style brainteasersSales and trading
Try it first
What comes next?
Show the worked solution
42. The gaps between terms are 4, 6, 8 and 10, rising by 2 each time, so the next gap is 12 and 30 + 12 = 42. The rule underneath is that the nth term is n x (n + 1): 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, and next 6 x 7, which is 42.
What should you do first with any number series?
Imagine a taxi meter that charges a little more for each extra kilometre than it did for the one before. The fares look irregular, but the jumps between them tell the story. Write the differences between terms underneath the series before guessing, and if they are not constant, take differences again. Here the first differences are 4, 6, 8 and 10, and the second differences are a steady 2, which tells you the rule is a quadratic: something times something.
The differences between 2, 6, 12, 20 and 30 are 4, 6, 8 and 10, rising by a steady 2, so the next difference is 12 and the next term is 42, which is also 6 x 7 under the rule n x (n + 1).
How do you check the answer, not just find it?
A second route that lands on the same number is your proof. Each term splits into two neighbouring whole numbers: 2 = 1 x 2, 6 = 2 x 3, 12 = 3 x 4, 20 = 4 x 5, 30 = 5 x 6. Two methods agreeing, differences and a formula, turn a guess into an answer. The formula also lets you jump ahead: the 10th term is 10 x 11 = 110 without writing out the terms in between.
The relationship
$a_n = n(n+1) \qquad a_6 = 6 \times 7 = 42$
a_n the nth term of the series
n its position, starting at 1
What it says in wordsEach term is its position multiplied by the next position.
Why does a bank put this in an online test?
Numerical and logical screens are timed tightly and filter large numbers of applicants before anyone reads a CV. The skill being tested is spotting structure fast, the same skill you use when a line in a model grows by a changing amount each year. A revenue line whose yearly increase itself rises by a fixed amount has exactly this shape, and seeing it saves you from projecting the last increase forward as if it were fixed.
Where candidates lose it
The fast wrong answer is 40, from assuming the last gap of 10 repeats. Under time pressure candidates spot the first differences and stop before noticing that the differences are themselves growing.
The other loss is spending a minute hunting for an exotic rule. Differences first, second differences next, then neighbouring products: those three checks crack most test series inside the time allowed.
What the interviewer asks next
What is the 20th term of the series?
What comes next in 1, 3, 6, 10, 15, and how is that series related to this one?
What is the sum of the first five terms, and is there a shortcut?

r-bar	the arithmetic average return, 10%
sigma	how far returns swing around the average, 20 points for Fund A

2^10	the number of different ill-or-well patterns 10 testers can show
0101100101	bottle 357 in binary, one digit per tester

0.02 x 0.95	the share of all companies that are frauds and get flagged
0.98 x 0.05	the share that are clean and get flagged anyway

H	households, about 30 crore
p	the share of households with a two-wheeler, 50%
L	the vehicle's life, 10 years
g	growth in the fleet, 3% a year

Try it first

Why does the same average give different endings?

How big is the drag, and can you estimate it in your head?

Why would an interviewer care?

Where candidates lose it

What the interviewer asks next

Try it first

What does each structure actually fix?

How do the two structures play out across the whole deal?

Which side wants which, and how do bankers split the difference?

Where candidates lose it

What the interviewer asks next

Try it first

Why does giving each tester a separate batch fail?

How does the binary code name the bottle?

What is the general lesson?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not 95%?

How do you work it with whole numbers?

What moves the number, and what does that mean for a screen?

Where candidates lose it

What the interviewer asks next

Try it first

Where do you start, and what do you refuse to guess?

How does the bottom-up build go?

Why build a second route, and what if it disagrees?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the rule to compare against 3.5?

How does the 4.25 come together?

What happens with a third roll, and why do interviewers ask?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a purchase on credit touch neither cash nor profit?

What happens over the next 60 days and beyond?

Where candidates lose it

What the interviewer asks next

Try it first

How do you count one size without listing every square?

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

Why is 1,296 the wrong answer here?

Where candidates lose it

What the interviewer asks next

Try it first

Why is it a weighted average and not a plain one?

Why does debt go in after tax?

Where candidates lose it

What the interviewer asks next

Try it first

What should you do first with any number series?

How do you check the answer, not just find it?

Why does a bank put this in an online test?

Where candidates lose it

What the interviewer asks next