Private Equity puzzles, solved step by step

He proposes 98 for himself, 0, 1, 0 and 1. Solve from the end. With two left, partner 4 takes everything. With three, partner 3 buys partner 5 for one unit. With four, partner 2 buys partner 4 for one. With five, partner 1 needs two votes besides his own and buys the two partners who would get nothing in the next round, 3 and 5, for one unit each.

Why start from the end rather than the beginning?

Think of planning a train journey with a fixed arrival time: you work back from when you must arrive to when you must leave. Each partner's vote depends on what they would get if the current proposer were removed, so you can only value a vote once you know the next round, and the only round you can solve directly is the last one. With two partners left, partner 4 proposes 100 for himself and 0 for partner 5. His own vote is one of two, which is half, and half is not fewer than half, so it passes.

With three left, partner 3 needs two yes votes. Partner 5 gets nothing if partner 3 is removed, so one unit buys him: 99, 0, 1. With four left, partner 2 needs two votes. Partner 4 gets nothing in the three-partner round, so one unit buys him: 99, 0, 1, 0.

Which votes does the most senior partner buy, and why so cheaply?

With five partners he needs three yes votes: his own and two more. Look at the four-partner row. Partners 3 and 5 get nothing there, while partner 4 gets one unit and partner 2 gets 99. The cheapest votes always belong to whoever would be worst off in the next round, and here that is partners 3 and 5, so one unit each beats their alternative of zero. He keeps 100 less 2, which is 98. The answer feels unfair, and that is the point: in this game power comes from position in the sequence, not from any idea of a fair share.

Which assumptions does the answer rest on?

Three, and naming them is part of the answer. The tie rule, the tie-breaking preference and pure self-interest each change the split if you alter them. If a plan needed a strict majority, the proposer would need more votes. If an indifferent partner voted yes, the bought votes would cost zero instead of one, and the senior partner would keep 100. And real partners care about fairness and future rounds of carry, which is why actual carry allocations are set by negotiation and track record rather than by this logic.

Ask either guide: which road would the other guide say leads to the site? Then take the other road. If you asked the truth-teller, he truthfully reports the liar's wrong answer. If you asked the liar, he lies about the truth-teller's right answer, which is also the wrong road. Either way the answer is the wrong road, so the right road is the one not named. The question works because it passes through both guides, so exactly one lie is applied.

Why does a direct question fail?

Imagine asking two shopkeepers whether a price is fair, knowing one of them always exaggerates, but not which. One answer is honest and one is not, and with no way to tell them apart you have learned nothing you can act on. A direct question about the road gives the truth from one guide and a lie from the other, and since you do not know which guide you asked, the answer carries no usable information. Asking a guide whether he is the truth-teller is worse: both say yes, the honest one because it is true and the liar because it is false. The trick has to work the same way whichever guide you happen to be facing.

How does routing the question through the other guide fix it?

By making sure the answer passes through exactly one liar. Ask the truth-teller what the liar would say and you get an honest report of a lie; ask the liar what the truth-teller would say and you get a lie about the truth; both contain one lie, so both point to the wrong road. The table in the figure checks all four cases: truth-teller or liar, left road or right road correct, and in every cell the named road is the wrong one. Because the answer is wrong by construction, you do not need to know which guide you asked. You simply take the other road.

Why does a deal desk ask a puzzle with no numbers in it?

Because it tests whether you can design a question whose answer is useful regardless of what you do not know. The puzzle is a small model of diligence: a management team, a seller's banker and a reference customer each have a reason to shade the truth, and the skill is to ask questions whose answers you can act on without first knowing who is being straight with you. Asking a supplier how the target compares with its other customers, rather than asking management how the supplier feels about them, is the same move: route the question so the bias cancels. The limit is that the puzzle assumes a guide who lies perfectly and consistently; a real counterparty who is merely selective needs several questions, not one.

Light rope A at both ends and rope B at one end, at the same moment. Rope A burns out after exactly 30 minutes, because two flames together consume 60 minutes of burning in half the time. At that moment rope B has 30 minutes of burning left, so light its other end. It burns out 15 minutes later, at 45 minutes in total.

Why does lighting both ends give exactly 30 minutes?

Two people painting a fence from opposite ends finish in half the time, even if some planks take longer than others, because they always work together on whatever is left. A rope holds 60 minutes of burning in total; two flames each burn their own share, and between them they use it up in 30 minutes, wherever they happen to meet. The uneven burn only changes where the flames meet, not when.

Why does rope B have exactly 30 minutes left?

Because time, not length, is what the rope stores. Rope B has burned for 30 minutes at the moment rope A goes out, so whatever length remains holds exactly 30 minutes of burning, and two flames finish that in 15. You never need to know where on the rope the flame is. That is the general lesson: when a quantity is uneven in one dimension, measure it in the dimension that is reliable.

Say it in order, with the two events that trigger each action: at the start, light three ends; when A goes out, light B's last end; when B goes out, 45 minutes have passed. Interviewers mark you on the clean sequence as much as on the idea. The limitation of the puzzle is its assumption that both ends can be lit at exactly the same moment.

Two weighings. Put three bars on each pan and three aside. If one pan rises, the light bar is among those three; if they balance, it is among the three aside. Then take that group of three and weigh one bar against one: the pan that rises holds it, and a balance means it is the bar left out. Each weighing has three outcomes, so two weighings tell nine bars apart.

Why split into three groups and not two?

Think of a balance scale as a question with three possible answers: left is lighter, right is lighter, or they balance. Asking a yes or no question wastes one of those answers. A weighing that splits the suspects into three equal groups uses every outcome, so each weighing cuts the suspects to a third rather than a half. Nine bars become three after one weighing and one after two.

How do you prove that one weighing is not enough?

Count the answers a single weighing can give: three. There are nine bars, so nine different possible answers to the question which bar is light. One weighing can distinguish at most three cases, two weighings at most nine, so with nine bars two is both enough and the minimum. The same count gives the general rule: n weighings can find one light bar among up to 3 to the power n bars, so three weighings handle 27.

Ten lockers are open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100, the perfect squares. Locker n is toggled once by each person whose number divides n, so its final state depends on how many divisors n has. Divisors come in pairs that multiply to n, which makes the count even, except for a perfect square, where one divisor pairs with itself. An odd count leaves the locker open.

What decides whether one locker ends open?

Think of a light switch flicked by everyone who walks past: if an odd number of people flick it, the light ends on. Locker n is flicked by person d exactly when d divides n, so the number of toggles is the number of divisors of n, and the locker ends open only if that number is odd. The whole puzzle reduces to one question: which numbers have an odd number of divisors? Do not simulate a hundred people; reason about one locker.

Why do only perfect squares have an odd number of divisors?

Pair each divisor with its partner: for 12, 1 goes with 12, 2 with 6, 3 with 4. Every divisor has a different partner, so the count is even and the locker closes. For a perfect square, one divisor is its own partner, 6 x 6 for 36, so it is counted once and the total turns odd. There are ten perfect squares from 1 to 100, so ten lockers stay open. The answer for any number of lockers N is the whole part of the square root of N.

The ball costs Rs 5 and the bat Rs 105. Call the ball x. The bat is x plus 100, so together they are 2x plus 100, which must equal 110. That makes 2x equal to 10 and x equal to 5. The fast answer of Rs 10 fails the check: a Rs 100 bat is only Rs 90 more than a Rs 10 ball.

Why does Rs 10 jump out, and why is it wrong?

The numbers are built so that 110 minus 100 hands you 10 without thinking. That subtraction answers a different question: what is left if the bat costs exactly 100. The condition is a difference, the bat is 100 more than the ball, not a price, so the bat must contain a whole ball's worth plus 100. Check Rs 10 against the condition and it fails at once: 100 minus 10 is 90.

What is the interviewer actually testing?

Not algebra. They are testing whether you check a fast answer against the conditions before you say it. A deal model full of quick numbers has the same risk: a cell that looks right because it is round, never tested against the constraint it was meant to meet. Writing one line, x plus x plus 100 equals 110, takes three seconds and removes the risk.

Say the answer, then the check in the same breath: ball 5, bat 105, difference 100, total 110. Interviewers often use this as a warm-up and judge you more on the check than on the number.