Mutual Fund Mastery puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 29
- Topics
- 13
- Hard
- 30
039Give an equation for the surface area of an n by n by n Rubik's cube, counted in small unit squares. Then give an equation for how many of the small cubes show at least one face.T. Rowe PriceBaltimore · 2020
Try it first
For a standard 3 by 3 by 3 cube, how many small cubes show at least one face?
Show the worked solution
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.
On a 4 by 4 by 4 cube the skin carries 96 squares, but only 56 cubes show, because each corner cube shows three squares and each edge cube two; the dashed 2 by 2 by 2 core of 8 cubes is hidden, and 64 minus 8 is 56. 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 relationshipn small cubes along one edge (n-2)^3 the hidden core after peeling one layer from every side V(n) cubes showing at least one face What it says in wordsVisible cubes are all the cubes minus the core you cannot see; expanded, it is the surface area less a correction for corners and edges.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.
Where candidates lose it
The trap is giving 6n² as the answer to both questions. The interviewer is checking whether you notice that one cube can show up to three squares. Name the two counts before you write anything.
The second loss is trying to count visible cubes from the surface and getting tangled in double counts at the edges. Go to the hidden core first, then offer the corner, edge and face breakdown as the check.
What the interviewer asks next
- How many small cubes show exactly two faces, as a formula in n?
- For which n is the hidden core larger than the visible shell?
- How would the counts change for a 4 by 5 by 6 box?
Asked at T. Rowe Price, Equities, Baltimore, 2020 (Wall Street Oasis):
Give an equation that yields the surface area of an n by n by n Rubic's cube based on number of blocks per side
088Five funds, A to E, are ranked by one-year return with no ties. C beat A but trailed E. A was neither first nor last. B finished immediately behind A. B beat D. What is the order from first to last, and which of the clues did you not actually need?Indian AMCsGlobal asset managers
Try it first
Which fund finished first?
Show the worked solution
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.
Joining the three ordering clues makes one chain of all five funds, E ahead of C ahead of A ahead of B ahead of D, so the positions are fixed before the clue about A's position or the word immediately is ever used. 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.
Where candidates lose it
The common loss is starting with 'A was neither first nor last', trying A in second, third and fourth place, and running a case for each. It works eventually, but it is slow and looks like guesswork, and under time pressure candidates drop a case and give a wrong order.
The second loss is giving the right order without proving it is the only one. Say the chain touches all five funds, so nothing can move, then name the two clues you never used.
What the interviewer asks next
- Remove the clue that B beat D. How many orders are now possible?
- Which single clue, if removed, would leave the order unchanged and still fully determined?
- Six funds now, with F known only to have beaten C. Where can F go?
