Venture Capital puzzles, solved step by step

Light fuse A at both ends and fuse B at one end, at the same moment. Fuse A burns out after exactly 30 minutes, because two flames use up its hour twice as fast. At that instant fuse B has 30 minutes of burning left, so light its other end too. It goes out 15 minutes later, at exactly 45 minutes.

Why does lighting both ends give exactly 30 minutes?

Think of two people eating from opposite ends of a plate of biryani that one person would finish in an hour. One may eat faster on the rice and slower on the meat, but together they always clear the plate in half the time. A fuse holds a fixed amount of burning time, 60 minutes, and two flames consume it at twice the rate of one, so they meet after 30 minutes wherever along the fuse that happens to be. The unevenness only moves the meeting point.

Why light fuse B at the start rather than later?

Fuse A is your clock for the first 30 minutes, and fuse B needs to spend those same 30 minutes burning so that exactly half its burn time is left. The trick is that fuse B stores time: after 30 minutes on one flame it holds exactly 30 minutes of burn, whatever its length, and a second flame halves that to 15. Lighting B only when A goes out would leave it with a full hour, and you could only measure 60 or 90 minutes from there.

What is the interviewer actually listening for?

Whether you separate what you know from what you do not. You do not know the burn rate along the fuse, so any plan that cuts or measures it by length is dead. You do know the total time, and the answer uses nothing else. That habit carries over to diligence: when a company's monthly numbers are noisy, lean on the totals you can verify rather than on a rate you are guessing.

190 one-to-ones in all, and 150 of those pairs never meet if talk stays within tables. Each of 20 founders pairs with 19 others, 380 counted from both sides, so 190 pairs. A table of five holds 5 x 4 / 2 = 10 pairs, and four tables hold 40. That leaves 190 minus 40, or 150 pairs, about 79% of all possible meetings, unmet.

Why do you divide by two?

Think of handshakes at a wedding. If you asked every guest how many hands they shook and added up the answers, each handshake would be counted twice, once by each person in it. A pair is one meeting seen from two sides, so the number of pairs among n people is n times (n minus 1), halved. For twenty founders that is 20 x 19 / 2, which is 190.

How many pairs does the seating leave out, and why is it more than three quarters?

Count what the tables allow, then subtract. A table of five holds 10 pairs, so four tables hold 40. Splitting a group into four tables does not cut the meetings to a quarter; it cuts them to about a fifth, because pairs grow with the square of group size. 40 out of 190 is 21%, so 150 pairs never meet. Check it from one founder's seat: each founder meets 4 people and misses 15, and 20 x 15 / 2 is also 150.

Why would a venture interviewer ask about pairs?

Pair counting sits under network effects. A marketplace with n users has about n squared over two possible connections, which is why tripling users multiplies the possible connections roughly ninefold. It is also why platform teams at funds run founder dinners in rotating rounds: moving people between tables is the cheapest way to raise the share of pairs that meet. Say the limit as well: a possible connection is not a valuable one, and most of the 190 pairs would have little to discuss.

Ask either founder, 'Which room would your co-founder say is real?', then take the other room. If you asked the truth-teller, she honestly reports the liar's false answer. If you asked the liar, he falsely reports the truth-teller's true answer. Either way the answer has passed through exactly one lie, so it always names the fake room. You do not need to know who answered.

Why does a direct question fail?

If you ask a stranger for directions and know only that they are either always honest or always lying, their answer alone is worth nothing: you cannot tell which kind of answer you got. A direct question returns the truth from one founder and its opposite from the other, so without knowing who answered, the reply carries no information. 'Are you the truth-teller?' is worse: both say yes. The question has to be built so the answer comes out the same whoever gives it.

Why does routing the question through the other founder fix it?

Count the lies along each path. Truth then lie, or lie then truth, both contain exactly one lie, so the indirect question always returns a false answer, and a reliably false answer is as useful as a true one: you simply take the opposite. There is a second classic: 'If I asked you whether room 1 is real, would you say yes?' The truth-teller answers honestly; the liar would say no to the inner question and then lies about that, so he says yes. That version uses two lies or none, which gives the truth directly.

The limit is the setup itself. The trick relies on people who lie with perfect consistency, which nobody does; real diligence on two founders who tell different stories relies on documents, not on clever questions. What the puzzle tests is whether you can design one check whose answer does not depend on who answers it, which is the logic behind cross-referencing a claim against two sources with opposite incentives.

Split the envelopes into three groups of three and weigh two of the groups against each other. The heavier pan holds the cheque; if they balance, it is in the group left aside. Then weigh two envelopes from that group: the heavier one is it, and a balance means the third. Each weighing has three outcomes, so two weighings tell apart 3 x 3 = 9 envelopes.

Why is a balance scale worth three answers, not two?

A traffic signal carries three messages, not two, because amber counts. A balance scale is the same: left heavy, right heavy, or level. The balanced outcome is information too, so the best weighing splits the suspects into three equal groups, two on the scale and one off it. Most people instinctively halve, as they would with a yes or no question, and halving throws away the third answer.

How do you prove two weighings are enough and one is not?

Count end points. With k weighings and three outcomes each, the scale can produce at most 3 to the power k different results, and each envelope needs its own result. One weighing gives three results, too few for nine envelopes. Two give nine, exactly enough, and the tree in the figure shows a plan that uses all nine. Three weighings would handle 27 envelopes the same way.

What does the interviewer want to hear beyond the procedure?

Say the counting argument, not just the steps. It shows you can tell when a plan is optimal rather than merely working. The habit carries over to diligence: a good question is one whose every answer, including the dull one, changes what you do next. A reference call where only a glowing answer would move you is a weighing that wastes the balanced outcome. Then state the assumption the puzzle rests on: you know the odd envelope is heavier, not merely different.