Private Equity puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 22
- Topics
- 12
- Hard
- 30
025At a fork in the road stand two guides. One always lies and one always tells the truth, and you cannot tell which is which. You may ask one guide a single question. What do you ask to find the road to the deal site?Mid-market buyout fund
Try it first
Which single question works?
Show the worked solution
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.
Whichever guide you ask and whichever road is correct, the answer to the question of what the other guide would say is always the wrong road, so taking the road not named is right in all four cases. 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.
You ask What he does Road he names What you do The truth-teller Reports the liar's answer honestly The wrong road Take the other The liar Lies about the truth-teller's answer The wrong road Take the other Both guides name the wrong road when asked what the other would say, so the rule is the same whichever one you met. 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.
Where candidates lose it
The common loss is panicking and asking the direct question, or asking a guide about himself, which both guides answer identically. Candidates also sometimes find the right question but then take the road it names, forgetting the final flip.
The second loss is giving the answer as a memorised line without showing it works in every case. Walk the four cases out loud, or draw the table, so the interviewer sees that you checked rather than recalled.
What the interviewer asks next
- What question works if there are three roads instead of two?
- Suppose the guides answer only yes or no. Rephrase the question so it still works.
- What question would tell you which guide is the liar, and why is that less useful than finding the road?
050You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn, but they burn unevenly, so half a rope's length does not take 30 minutes. How do you measure exactly 45 minutes?Large-cap buyout fund
Try it first
What is the key move?
Show the worked solution
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.
Rope A, lit at both ends, burns out at 30 minutes whatever its burn rate; rope B, lit at one end, then has exactly 30 minutes of burning left, and lighting its other end at that moment makes it last 15 more, so it goes out at 45. 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.
Where candidates lose it
The common loss is trying to use length: cutting a rope in half or marking three quarters of it. The question says the burn is uneven precisely to rule that out, and candidates who miss the word lose the room.
The second loss is lighting rope B's second end at the start as well, which gives 30 minutes, not 45. B must burn from one end for the first 30 minutes.
What the interviewer asks next
- With the same two ropes, how would you measure 15 minutes?
- What times between 0 and 120 minutes can you measure with two such ropes?
- How many ropes would you need to measure 7.5 minutes?