Financial Analysis puzzles, solved step by step

Weigh coins 1 to 4 against 5 to 8. If they balance, the odd coin is among 9 to 12 and two weighings against known good coins find it. If one side is heavy, eight cases remain, and a second weighing that swaps some coins across the pans cuts them to three or fewer. It works because three weighings give 27 outcomes and there are only 24 cases.

Why can three weighings be enough?

Picture a guessing game where every answer is higher, lower or spot on. A question that leaves you with one of three equal groups does three times the work of a yes or no. A balance has three outcomes, left heavy, right heavy or level, so three weighings can tell apart 3 x 3 x 3 = 27 situations, and this puzzle has only 24: any of 12 coins, heavy or light. The plan works only if every weighing splits the remaining cases as evenly as possible into three.

That rules out six against six: one side always tips, so the balance outcome is wasted and twelve cases are left for two weighings that can separate nine. Four against four is right. If the left pan is heavy, the odd coin is one of 1 to 4 and heavy, or one of 5 to 8 and light: eight cases. If the pans balance, coins 1 to 8 are good and the odd coin is among 9 to 12, heavy or light: eight cases again.

How does the unbalanced branch get from eight cases to three?

Suppose the left pan was heavy. Weigh 1, 2, 5 against 3, 6 and coin 9, which is known to be good. Coins 4, 7 and 8 sit out. If the left is heavy again, the culprit is 1 or 2 heavy, or 6 light; weigh 1 against 2, and if they balance it is 6. If the right is now heavy, it is 3 heavy or 5 light; weigh 3 against a good coin. If they balance, it is 4 heavy or 7 or 8 light; weigh 7 against 8, and the lighter one is the culprit, or 4 if they balance. The trick is that some coins stay, some swap pans and some leave, so each outcome points to a different group.

If the first weighing balanced, weigh 9, 10, 11 against good coins 1, 2, 3. A heavy left means one of 9 to 11 is heavy, and 9 against 10 finds it. A light left means one is light, found the same way. A balance means coin 12 is odd, and weighing it against any good coin says whether it is heavy or light.

Why would a finance interviewer ask this?

It tests whether you count the possibilities before you start moving coins, which is the habit behind good diligence: design each question so that every possible answer tells you something new. The interviewer is happy with a clear first move and the counting argument, even if you need a minute for the unbalanced branch. The limit worth adding: with thirteen coins you can still find the odd coin in three weighings, but not always say whether it is heavy or light.

14% today, rising to about 16.7% after B doubles, with no change in either division's own margin. Group EBITDA is 80 + 60 = Rs 140 crore on Rs 1,000 crore. When B grows to Rs 400 crore at 30%, EBITDA is 80 + 120 = Rs 200 crore on Rs 1,200 crore. The group margin is a revenue-weighted average, so it moves when the weights move.

Why is the group margin not the average of the two?

A cafe earns 30% on coffee and 10% on sandwiches. If it sells mostly sandwiches, its overall margin sits near 10%; if coffee takes off, the overall margin climbs, even though neither item is any more profitable. A group margin is a revenue-weighted average of segment margins, so it moves whenever the mix moves, even if no segment changes. Today B is a fifth of revenue, so the group sits a fifth of the way from 10% to 30%: 14%.

Can the group margin rise while every division gets worse?

Yes. Suppose that while B doubled, A's margin slipped to 9% and B's to 28%. Group EBITDA would be 72 + 112 = Rs 184 crore on Rs 1,200 crore, 15.3%, still above the 14% start. Both divisions got less profitable and the group margin still rose, because the shift towards B outweighed the decline inside each. Statisticians call this pattern Simpson's paradox, and a management commentary that cites only the group margin can hide it.

What should an analyst do with a reported margin gain?

Split it into mix and rate. The mix effect is the margin you would get with the new weights and the old divisional margins, less the starting margin: 16.67% less 14% is 2.67 points. The rate effect is the rest. In the puzzle the rate effect is zero; in the deteriorating version it is -1.33 points, which is the story a reader needs. The limit: segment data is often reported only twice a year and with shared costs allocated by management, so the split is only as clean as the allocation.

Root 50 is about 7.07, and 1.2% a day is about 19% a year. Root 49 is 7, and near a known square the curve is nearly straight, so add the step divided by twice the root: 7 + 1 / 14 = 7.0714 against a true 7.0711. Volatility scales with the square root of time, so multiply by root 250, about 15.81: 1.2 x 15.81 = 19.0%.

How do you get root 50 without a calculator?

Think of walking up a gentle hill: one more step forward lifts you by the slope at the point you are standing, and for a single step the hill may as well be straight. Near a square you know, the root curve is nearly straight, so the root of 49 plus 1 is 7 plus 1 divided by the slope's denominator, twice 7. That gives 7 + 1/14 = 7.0714; the true value is 7.0711, an error of 0.0004. The slope of the root curve at a point is one over twice the root, which is why the denominator is 14 and not 7.

Why does volatility scale with the square root of time?

Toss a coin 100 times and count heads minus tails. The lead is almost never 100; it is typically about 10, the root of 100, because the tosses partly cancel. Independent moves add in variance, not in standard deviation, so the spread after n days is the daily spread times root n. Daily variance is 1.2 squared, 1.44; over 250 days it is 360; the root of 360 is 19.0. Root 250 by the tangent at 256 is 16 - 6/32 = 15.812, close enough to the true 15.811.

When does the root-of-time rule mislead?

The rule assumes each day's move is independent of the last and drawn from the same distribution. Trending markets, where moves follow moves, make the true spread wider than root n suggests; mean-reverting ones make it narrower. Volatility also clusters, so a 1.2% daily figure measured in a calm month understates a stormy one. The day count is a convention: 250, 252 and 365 are all in use, and annualised volatilityThe standard deviation of returns restated to a one-year horizon, usually by multiplying a daily figure by the square root of the number of trading days. quoted on each will differ, so state the one you used. The limit to say: the rule converts a horizon, it does not forecast the year.

Play: one roll pays 30 x 3.5 = Rs 105 on average against a Rs 100 fee, an edge of Rs 5. With the re-roll, keep a 3 or better and re-roll a 1 or 2; the game is worth Rs 118.33, an edge of Rs 18.33. A fresh roll is worth 105 less the Rs 20 fee, 85, so re-roll only a result paying less than 85: 30 and 60 qualify, 90 does not.

Why is the first game worth playing?

A fair die averages 3.5, so the average payout is 30 x 3.5 = Rs 105, more than the Rs 100 you pay. On expected value the game pays Rs 5 a play, so a player who can repeat it should play. Half the time you lose money, since a 1, 2 or 3 pays 30, 60 or 90, and the other half you win 120, 150 or 180; the wins are larger than the losses, which is where the edge sits.

How does the re-roll change the rule?

Think of returning a shirt. You keep it if what you have is worth more than what the shop will hand you after the restocking fee. Re-roll only when the roll in hand is worth less than a fresh roll net of its fee, which is 105 - 20 = Rs 85. A 1 pays 30 and a 2 pays 60, both below 85: re-roll. A 3 pays 90, above 85: keep it, even though 3 is below the average face. The fee is what makes a 3 worth keeping.

What does the option itself cost and earn?

The re-roll lifts the game from 105 to 118.33, so the option is worth Rs 13.33 on average, and it also cuts the chance of losing money from a half to a third, because a bad first roll gets a second chance. Re-rolling a 3 as well would give 117.50, worse by 0.83; a free re-roll would make re-rolling the 3 right and lift the game to 127.5. The threshold moves with the fee: below a fee of Rs 15 the 3 is worth re-rolling, above it the 3 is kept. The limit to say: the worst path, a 1 or 2 followed by a 1, costs Rs 90, and a player who cannot afford that should not value the game by its average.

From shrinking sales, not better collecting. With days sales outstanding fixed at 73, receivables are a fifth of revenue, so they fell from Rs 200 crore to Rs 150 crore as revenue fell Rs 250 crore, releasing Rs 50 crore. Unchanged days means customers paid no faster. The release is a one-off that reverses when sales recover, and it was bought with a quarter of the revenue.

Why does falling revenue release cash?

A tailor who gives customers two months to pay always has about two months of sales outstanding. If orders halve, the amount owed to him halves too, and for a while he collects old bills faster than he issues new ones: cash arrives. Receivables are revenue times the collection period, so when the period is fixed, receivables move with revenue, and a fall in sales releases cash once, mechanically. Here 73 days is 73 / 365, one fifth of a year, so receivables are a fifth of revenue: 200 on 1,000, 150 on 750. The Rs 50 crore is a fifth of the Rs 250 crore of sales that disappeared.

How do you separate the volume effect from the collections effect?

Split the change in receivables into two pieces. The volume effect is the change in revenue times the old days: (750 - 1,000) x 73 / 365 = -50. The collections effect is the new revenue times the change in days: 750 x (73 - 73) / 365 = 0. Every rupee of the release is volume; the collections effect is exactly zero, and that is the number management's claim rests on. Had days improved to 60, receivables would be Rs 123 crore, a release of 76.7: 50 of volume and 26.7 of genuine improvement, and only that 26.7 would deserve the word collections.

What does the analyst say about the quality of the Rs 50 crore?

Three things. It is one-off: receivables cannot keep falling unless sales keep falling. It reverses: if revenue climbs back to Rs 1,000 crore at 73 days, receivables return to 200 and the Rs 50 crore is reabsorbed. And it was expensive: at a 30% contribution margin, Rs 250 crore of lost revenue costs about Rs 75 crore of contribution every year, more than the cash released once. The limit: this uses year-end revenue and year-end receivables, and a company whose sales fell late in the year can show a days figure that flatters or punishes it; check the quarterly pattern before concluding.

Lifetime value is about Rs 11,667, LTV to CAC is about 1.9x, and payback is about 17 months. Each customer brings Rs 350 of gross profit a month, 70% of Rs 500. At 3% churn the average customer stays 1 / 0.03 = 33.3 months, so LTV is Rs 350 x 33.3. The Rs 6,000 acquisition cost divided by Rs 350 a month is paid back in 17.1 months.

Why is the average customer life one over churn?

Think of a hostel where 3 residents in every 100 move out each month and are replaced. A resident faces a 3% chance of leaving every month, so on average the wait before leaving is 1 / 0.03 months. With a constant monthly churn rate, the expected customer life is one divided by that rate: at 3%, 33.3 months. Lifetime value is the gross profit the customer brings each month times that life.

Use gross profit, not revenue. The Rs 150 a month it costs to serve the customer is spent whether or not the customer was worth acquiring, so it cannot pay back the acquisition cost. Rs 350 x 33.33 months is Rs 11,667, and against a CACCustomer acquisition cost: marketing, sales and onboarding spend divided by the number of customers it won. of Rs 6,000 that is 1.94x.

Why are there two payback answers?

Payback is the month in which cumulative gross profit covers the Rs 6,000 spent on day zero. For a customer who stays, that is 6,000 / 350 = 17.1 months, the standard answer. Across a whole cohort payback is slower, because some customers leave before they have repaid their share of the acquisition cost. Weight each month by the chance the customer is still there and the average customer crosses Rs 6,000 in month 23.7. Give 17 months and name the cohort figure in one sentence.

Is 1.9x good enough?

Many subscription investors use a rule of thumb of 3x or better with payback inside about a year; treat those as conventions, not laws. At 1.9x this business recovers its acquisition cost with a thin cushion, and the ratio flatters it, because LTV here is undiscounted and ignores any cost of retaining customers. Churn is the strongest lever: cutting it from 3% to 2% lifts LTV by half to Rs 17,500, while a 10% price rise at the same churn only reaches Rs 12,833.

Not on this evidence: most of the drop is regression to the mean. A branch that tops the table usually had skill plus a good year. The good year does not repeat, so the same branches fall back toward the average whatever the manager does. Here only 2 points of a 20 point lead survived, so most of last year's lead was luck. Judge the head against similar branches that kept their managers, not against their own lucky year.

Why do the best performers fall back with no cause at all?

Think of a class test. The student who topped it probably knows the subject and also had a good day: the questions suited her and two guesses landed. In the next test the knowledge stays but the luck is drawn again, so her score is likely to be lower while staying above average. Any result that is part skill and part luck will, when it is extreme, usually be followed by a less extreme one. Francis Galton called this regression toward the mean when he compared the heights of parents and their children.

How much of the 30% was luck?

Measure the lead, not the level. The top five were 20 points above average last year and 2 points above this year. Keeping 2 points of a 20 point lead means about a tenth of last year's outperformance was persistent, and nine tenths was noise that happened to land on those five branches. Look at the bottom of the chart as well: the weakest five rose from -10.0% to 8.3% with no new manager, which is the same drift running the other way.

How would you test whether the new head made a difference?

Build a comparison. Take branches that were about as strong last year but kept their managers, and see how they grew this year. If they also fell to around 12%, the new head is doing exactly what chance predicts. If they held near 20%, the head has a real question to answer. The fair benchmark for a group picked for being extreme is a similar group picked the same way, never its own best year. The same trap sits in sales incentives, fund selection and bonus pools, where praising last year's winners and punishing last year's losers both appear to work.

The real raise is 3.77%, so the 4% shortcut overstates it by about a quarter of a point. Real growth divides growth factors: 1.10 / 1.06 = 1.0377. The shortcut subtracts the rates, which ignores that the raise is itself paid in money worth 6% less. At low rates the gap is small; at a 30% raise with 20% inflation the shortcut says 10% against a true 8.33%.

Why divide rather than subtract?

Measure your pay in lunches instead of rupees. Last year a thali cost Rs 100 and you earned Rs 100 a day: one thali. This year the thali costs Rs 106 and you earn Rs 110, so you can buy 110 / 106 = 1.0377 thalis. Your real pay rose 3.77%, because real growth is the ratio of two growth factors, not the difference between two rates. The relation is usually credited to the economist Irving Fisher.

The second form shows exactly where the shortcut goes wrong. The 4% gap is right in the numerator; it just has to be shrunk by dividing by 1.06. The error is small when inflation is small and grows as inflation rises.

When does the shortcut become a real error?

Three places. When rates are high, as the chart shows. When the rate compounds: over ten years, 4% a year grows purchasing power by 48.0% but the true 3.77% grows it by 44.8%, a gap of about three points. And when a budget is being judged: an FP&A analyst splitting revenue growth into price and volume uses exactly this division, and subtracting instead misstates the volume growth the business really delivered. For quick talk in a meeting, 4% is fine; in a model, divide.

12% a year at a 40% payout, and 6% at 70%. With no new equity and fixed leverage, the only fuel for growth is retained profit. Equity grows by ROE times the share of profit kept: 20% x 60% = 12%. Debt grows in step to hold the ratio, so assets, and at a steady asset turnover sales, can grow 12%. Keep only 30% and the ceiling halves to 6%.

Why is retained profit the only fuel?

A family shop that refuses outside partners and refuses to borrow more than it already does relative to its size can grow only on the profit it leaves in the till. With new equity ruled out and leverage fixed, equity can grow only by retained profit, and debt can grow only as fast as equity. So the whole balance sheet is capped at the rate equity compounds.

Put numbers on it. Opening equity of Rs 100 crore earns Rs 20 crore at a 20% ROE. Rs 8 crore is paid out and Rs 12 crore kept, so equity closes at Rs 112 crore, 12% higher. Next year's profit on Rs 112 crore is Rs 22.4 crore, also 12% higher, and the pattern repeats.

What assumptions are hiding inside the formula?

The formula, often taught through Robert Higgins' sustainable growth rate, assumes the 20% ROE holds on every new rupee of capital. Split ROE the DuPont way and it could be a 5% net margin x 2.0 asset turnover x 2.0 assets to equity. Growth above 12% therefore needs one of five things: a better margin, faster asset turnover, more leverage, new equity, or a lower payout. That list is the useful part of the answer, because it is exactly the set of levers a finance team argues about when a plan grows faster than its funding.

The limitation is worth one sentence: ROE rarely stays flat as a company grows, because new projects are usually less profitable than the best existing ones. The 12% is a ceiling under today's economics, not a forecast.

Light rope A at both ends and rope B at one end, together. When A burns out, 30 minutes have passed; light B's other end, and B finishes 15 minutes later, at 45 minutes. Lighting both ends halves any rope's remaining burn time whatever its speed profile, because two flames use the time up together. B has 30 minutes of burn left at the half hour, and two flames clear that in 15.

Why does lighting both ends give exactly half, when the rope burns unevenly?

Picture two friends eating one plate of biryani from opposite sides. One side is all rice and goes fast, the other is full of bones and goes slowly. They will meet somewhere off centre, but if the plate takes one person 60 minutes, two people eating at the same moment finish it in 30, because between them they are always doing two minutes of eating every minute. The rope's total burn time is fixed at 60 minutes; two flames consume it twice as fast, so it lasts 30 minutes regardless of where the slow sections are.

That is the whole trick. Measuring by length fails, which is why cutting a rope in half or marking its midpoint gets you nothing: the midpoint of the string is not the midpoint of the time. Measuring by burn time is exact.

Where do the last 15 minutes come from?

Rope B was lit at one end at the same moment as A. When A dies at minute 30, B has burned for 30 minutes, so it holds exactly 30 minutes of burn time, although nobody knows how long the remaining piece is. Lighting B's unlit end at that moment halves its remaining 30 minutes to 15, so B goes out at 30 plus 15, or 45 minutes. Both ropes are doing the same job: A is a 30 minute timer, and B is a 60 minute timer that you shorten halfway through.

Say the steps in order and say the start condition: all three ends must be lit at the same instant. An interviewer listening to a phone screen is checking that the sequence is clean and that you explain why it works, not just that you have heard the riddle before.

What else can two ropes measure?

With one rope you can time 60 or 30 minutes. With two you can add 15, 45, 75 and 90: 90 by burning one rope at both ends and then the other at one end, 75 by burning A at both ends, then B at one end for 30 minutes and lighting its second end. Each rope gives you only halving moves on whatever time it has left, so the times you can build are sums of 60, 30 and 15. You cannot get 20 minutes with these ropes, and saying why is a strong close.