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.

The CTO owns 50%, the COO 30% and the CEO 20%. Start with the COO, the person two clues mention. Not owning the least rules out 20. The CTO owning more rules out 50, since nobody can own more than 50. So the COO holds 30, the CTO must hold the 50, and the CEO has the 20 left, which fits the clue that the CEO does not own 30.

Where should you start a puzzle like this?

Seating a dinner table works the same way: you begin with the guest who has the most constraints, because each rule about them removes options for everyone else. Start with the person the most clues mention, here the COO, since two clues bear on the COO and each one deletes a column. Not owning the least removes 20. The CTO owning more than the COO removes 50, because there is no larger stake for the CTO to hold. That leaves the COO at 30 before you have used the third clue.

How do you know the answer is the only one?

There are only six ways to hand three stakes to three people, so you can check them all, but the grid makes it quicker. Once the COO is fixed at 30, the relational clue forces the CTO into the one stake larger than 30, and the last stake goes to the CEO by elimination. Then use the clue you have not used yet as a check: the CEO holds 20, not 30, so it is satisfied. A clue that was never needed to find the answer but still holds is good evidence you have not made a slip.

Say the method as you go. An interviewer giving a one-minute logic puzzle is listening for an order of reasoning, not just the answer, so name which clue you use at each step. The trap is to start with the CEO, because the CEO is listed first, and then guess. One clue about the CEO removes only one cell, so you are left trying cases.

Why would a venture interviewer ask a founders' puzzle?

It is a warm-up that tests clean reasoning under mild time pressure, dressed in the language of a cap table. It also hints at something real: founder splits rarely follow titles, and an investor reading a cap table should not assume the CEO holds the largest stake. The limitation is obvious: real ownership questions are settled by the share register and the shareholders' agreement, not by inference, and the useful skill here is the elimination habit, not the founders.

Ten tests, run at once. Number the statements and write each number in binary, ten digits long. Test pool 1 holds every statement whose first digit is 1, pool 2 every statement whose second digit is 1, and so on. A week later the ten results, read as positive equals 1, spell out the forged statement's number. Ten is the minimum because nine tests give only 512 possible result patterns, fewer than the 1,000 statements.

Why can you not just halve the pile and test again?

Halving is the right instinct when you can see each answer before asking the next question, like guessing a number with higher or lower clues. Here every result takes the full week, so all the testing has to be designed up front. When questions must be asked in parallel, each statement needs its own unique pattern of answers, and the puzzle becomes how many yes or no answers it takes to give 1,000 statements a distinct pattern each. Ten answers give 2 to the 10, or 1,024 patterns; that is enough.

How do you prove ten is the fewest?

Count the outcomes. Each test has two results, so n tests have at most 2 to the n distinct result patterns, and you need at least one pattern per possible forgery. Nine tests give 512 patterns for 1,000 suspects, so at least two statements would share a pattern and you could not tell them apart. Ten give 1,024, which is why 10 is both achievable and the minimum. One small detail: if you number the statements 1 to 1,000, the all-negative pattern is never used, which is fine because one statement is definitely forged.

Say where the trick breaks. It relies on exactly one forgery; with two forged statements, the positives are the union of two binary numbers and no longer name either one, so you would need more tests and a different design. It also assumes the test is perfectly sensitive in a pool of 500 statements. In real diligence a forger rarely fakes only one document, so the useful habit is the reasoning, counting outcomes before designing the checks.

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.