Mutual Fund Mastery puzzles, solved step by step

No. Tarvela's share fell from 8.0% to about 7.4%. Its AUM grew 30% while the industry grew 40%. Share is 40,000 over 5,00,000 before and 52,000 over 7,00,000 after. To hold 8% Tarvela needed Rs 56,000 crore, so it is Rs 4,000 crore short of standing still. Growing slower than your market means losing share while growing.

Why can a business grow and still shrink?

A child who grows 3 cm in a year while every classmate grows 5 cm is taller than last year and lower in the line-up. Market share is a ratio, so it rises only when your growth beats the market's, and a large rupee gain means nothing on its own. Tarvela's Rs 12,000 crore gain sounds big until you set it next to the industry's Rs 2 lakh crore.

What is the fastest way to answer without dividing big numbers?

Compare growth factors. New share equals old share times Tarvela's growth factor over the industry's: 8% times 1.30 over 1.40. 1.3 over 1.4 is about 0.93, so the share falls by about 7% of itself, from 8.0% to 7.43%. You never need to divide 52,000 by 7,00,000 unless asked for the decimal.

Add the analyst's question. AUM growth has two parts, market movement and net flows. If Tarvela is heavier in a segment that rose less, it could lose share with no problem in its sales. Splitting growth into the part the market gave and the part investors gave is the next thing an interviewer will ask for.

200 miles. The car covers 100 miles at 50 miles an hour, so the journey lasts two hours. The bird flies the whole time at 100 miles an hour, so it covers 200 miles however many times it turns. The long way agrees: the bird's round trips are 133.3, then 44.4, then 14.8 miles, each a third of the last, summing to 133.3 times 1.5, 200.

What is the question really asking?

Think of a dog running back and forth between you and your front door while you walk home. You could trace every turn, or you could notice the dog runs the whole time you are walking. When something moves at a constant speed for a known length of time, distance is speed times time, and the path it takes in between does not matter. The bird's speed is given; the only thing you need is how long it flies, and that is the car's travel time.

Can you prove it the long way, too?

Yes, and doing so briefly earns credit. The car and bird close a 100 mile gap at 150 miles an hour, meeting after two thirds of an hour, 66.7 miles from B. The bird flies 66.7 miles back to B, arriving at 1 hour 20 minutes, when the car is 66.7 miles from A. Each new chase starts with a gap a third of the last, so each round trip is a third as long: 133.3, 44.4, 14.8 and so on. A geometric series with ratio one third sums to the first term times 1.5, which is 200.

Why would a fund interviewer ask this? It tests whether you look for the quantity that is fixed before diving into detail, the same habit that makes you ask what an investor actually earned before reconciling every trade. The limitation is that it is a pure puzzle; turning instantly at each meeting is an idealisation that makes the infinite number of turns harmless.

The surface is 6n² unit squares, and n³ minus (n minus 2)³ cubes show at least one face. Six faces each carry n by n squares. For the cubes, count what is hidden: peeling one layer off every side leaves an (n minus 2) cube inside, so the visible ones are n³ minus (n minus 2)³, which expands to 6n² minus 12n plus 8. For a 3 by 3 by 3 cube that is 54 squares and 26 cubes.

Why are there two different counts here?

A house with a corner room has windows on two walls of that room, but it is still one room. Counting windows and counting rooms give different numbers. The surface counts squares, and a corner cube carries three squares and an edge cube two, so the square count is always larger than the number of cubes that show. Keeping the two counts apart is half the problem; candidates who blur them give 54 for both.

The surface is the easy part. Each of six faces is an n by n grid, so the area is 6n² unit squares: 54 for a standard cube. A quick check is to rebuild it from the cube types: 8 corners showing 3 squares, 12(n minus 2) edge cubes showing 2, and 6(n minus 2)² centre cubes showing 1. For n of 3 that is 24 plus 24 plus 6, which is 54 again.

Why count the hidden cubes instead of the visible ones?

The hidden cubes form one clean block, (n minus 2) on every side, so counting them and subtracting from n³ avoids every double count. Counting the visible cubes directly means adding corners, edges and faces while remembering that each edge already lost its corners. Both routes give the same answer, and the second makes a good check.

The expanded form is worth a sentence because it links the two answers. 6n² is the square count; the minus 12n plus 8 removes the extra squares that edge and corner cubes contribute. Say the edge case too: the formula needs n of at least 2. For n of 1 it gives 2, but a single cube is one cube.

It is a 0.3 percentage point cut, which is 30 basis points, and a 20% relative cut in the fee; "a 0.3% cut" is the sloppy one. Basis points remove the ambiguity. The 20% is true but invites a client to imagine returns rising 20%; on Rs 10 lakh the saving is Rs 3,000 a year, and a net return of 8.5% becomes 8.8%.

Why are there two different answers that are both right?

A shop raises the price of milk from Rs 50 to Rs 60. It went up by Rs 10, and it went up by 20%. Nobody confuses those, because rupees and percent look different. With rates, the change and the base are both percentages, so "percent" can mean the gap between two rates or the change relative to the old rate, and you must say which. The gap between 1.5% and 1.2% is 0.3 percentage pointsThe plain difference between two percentages. From 1.5% to 1.2% is a fall of 0.3 percentage points.. Relative to the old fee, 0.3 is a fifth of 1.5, a 20% cut.

Which description will a client get wrong, and what should you say instead?

"A 0.3% cut" fails one way: a careful reader can take it as 0.3% of the fee, which would leave 1.5% x 0.997 = 1.4955%, a cut a hundred times too small. "A 20% cut" fails the other way: it is true of the fee, but a client hears 20% and imagines the investment doing 20% better. The fee is a slice of a slice; a 20% smaller slice lifts a net return of 8.5% to 8.8%, only 3.5% better in relative terms. The clean form for a desk is basis pointsHundredths of a percentage point. 30 basis points equal 0.30 percentage points.; the clean form for a client is rupees: Rs 15,000 a year on Rs 10 lakh becomes Rs 12,000.

The same confusion runs through everything a fund desk reports: a yield moving from 7.0% to 7.5% is 50 basis points, not 0.5%, and a fund outperforming by 2% might mean two points or a fiftieth more. The limit worth stating: neither unit tells the client the money, so pair any rate change with the rupee effect on their actual holding.

Weigh three coins against three. The pan that rises holds the light coin; if they balance, it is among the three left aside. Then weigh two of those three suspects against each other: the rising pan holds it, and a balance means it is the third. Each weighing has three possible outcomes, so two weighings can separate 3 x 3 = 9 coins.

Why split into three groups rather than two?

A shopkeeper asking a customer "yes, no, or not sure?" learns more from one answer than one asking only "yes or no?". A balance scale gives three answers, left pan rises, right pan rises, or the pans balance, so the best weighing splits the suspects into three equal groups and lets the scale tell you which group holds the light coin. Put coins 1 to 3 on the left and 4 to 6 on the right. If the left pan rises, the light coin is 1, 2 or 3. If the right pan rises, it is 4, 5 or 6. If they balance, it is 7, 8 or 9.

How do you know two weighings is the minimum, and how far does this go?

Count outcomes. One weighing has 3 outcomes, two have 3 x 3 = 9, and there are 9 possible light coins, so two weighings are just enough. The number of weighings needed is the smallest w with 3 to the power w at least the number of coins: 9 coins need 2, 27 coins need 3. One weighing can only handle 3 coins, so you cannot do 9 in one. Splitting into halves wastes the balance outcome and needs three weighings for nine coins.

The desk lesson is about information, not coins: a test with three outcomes is worth more than one with two, if you design it to use all three. The limit of the puzzle is that you are told the odd coin is lighter. If it could be heavier or lighter, each coin has two possible states, there are more cases to separate, and the strategy needs more care, which is the classic twelve-coin follow-up.

E, C, A, B, D, and two of the clues were not needed. C beat A but trailed E gives E ahead of C ahead of A. B behind A extends the chain, and B beat D adds D at the end: one line of five, so the order is fixed. A being neither first nor last, and B being immediately rather than merely behind A, are both already true of that chain.

Where do you start with a ranking puzzle?

Seating guests at a dinner, you place the couple who must sit together before the guest who will sit anywhere. Start with the clues that relate two items directly, and join them into chains; a chain that runs through every item fixes the whole order at once. Here every 'beat' clue shares a fund with another clue: E over C, C over A, A over B, B over D. Each clue hands the next one a fund to hang on, so the chain builds without any guessing.

How do you know the order is the only one?

Because the chain is complete. Every fund sits on one line of 'finished ahead of' links, so there is no fund whose position is free to move. When a chain touches all five items, uniqueness is proved, not hoped for. If any fund had hung off the chain, say D with only 'B beat D' and no other link, you would have had to test each place it could go. Checking uniqueness out loud is what separates a candidate who got lucky from one who reasoned.

Why would an interviewer include clues you do not need?

Two pieces of the puzzle are redundant. A being neither first nor last follows from A sitting third in the chain. And the word immediately adds nothing: B merely behind A is enough, because once E and C are ahead of A and D is behind B, there is no fund left that could sit between A and B. Spotting redundancy shows that you checked what each clue does rather than ticking them off. Interviewers also use extra clues to tempt you into starting from a weak one; A's 'neither first nor last' narrows A to three places, which feels like progress and fixes nothing.

The fund-desk version of the same habit is reading a factsheet. A list of statements about a fund, its rank, its category, its risk grade, often contains claims that follow from each other, and the analyst's job is to find the few that carry information.