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How to approach an interview question
The questions differ by subject and the method does not. This is what to do in the first minute, and what the person opposite is actually judging.
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Think aloud
Talk from the first second, including the ideas that turn out to be wrong. The interviewer is judging how you reason, not only where you end up, and silence gives them nothing to work with.
There is no need to narrate every pen stroke. Say what you are trying and why: "trying a small case first", "this should come out at about a thousand, so checking that". If you need a moment to think in silence, say so, and then come back with what you found.
When an idea fails, say why it failed. "That cannot be right, it gives a negative length" is good reasoning, and it often points to the next idea.
Start with small cases and special values
Most of these questions open up after one example. Before reaching for algebra, try n = 1, 2 and 3, put x = 0 or x = 1, or draw the simplest picture you can.
Small cases show the pattern, and they also give you something to check a general answer against. If your formula gives the wrong value for n = 2, you know at once.
Special values work in physics, engineering and chemistry too: what happens if the mass is zero, the angle is 90 degrees, or the rope is very long? The extreme cases are often easy, and the answer in between has to join them up.
Look for what stays the same
Many questions turn on a quantity that cannot change: energy, momentum, a total, the colour count on a chessboard, whether a sum is odd or even. Ask early on what is fixed and what is allowed to change.
Once you have found that quantity, the problem usually becomes much smaller. You no longer need to follow every step, only to compare the start with the end.
If nothing obvious stays the same, try combining things: a sum, a product, or one more than each number may be fixed even when each part moves.
Estimate with round numbers
Use numbers you can multiply in your head: the Earth's radius is about 6,400 kilometres, g is about 10, a person is about 70 kilograms, a mole is about 6 x 1023. Write powers of 10 rather than long strings of zeros.
State each assumption as you make it, so the interviewer can follow and correct you. Then say which assumption the answer is most sensitive to, and roughly how far off it could be.
An answer within a factor of 2 or 3 is usually a good estimate. Getting the power of 10 right matters far more than the first digit.
Use hints well
Interviewers often give hints, and taking one is not a failure. It is the normal shape of the conversation, and it shows them how you learn something new on the spot.
When a hint comes, pause, say what it changes, and carry on from there. If you do not understand it, ask a short question rather than nodding and hoping.
Do not hold on to a route that the interviewer is gently steering you away from. Leaving a dead end gracefully is a skill in itself.
Check the answer
Before you finish, test what you have. Does it have the right units? Is it about the right size? Does it behave sensibly at the extremes, when a quantity is very small or very large?
Try the answer on a small case you already worked out. For a probability, check that it lies between 0 and 1; for a physical quantity, compare it with something you know.
Saying these checks out loud is often worth as much as the answer itself. If a check fails, say so calmly and look for the mistake: finding your own error is a strength.
Practise in pairs
Work with a friend, a teacher or a parent. One asks, one answers aloud, and the asker gives the follow up only once the first part is done. Then swap.
The asker should hold back the hint for a while and offer it only when the answerer is truly stuck, as an interviewer would. Reading the hint too soon takes away the practice.
Aim for practice that feels like the real thing: about 10 to 15 minutes per question, talking the whole time, and writing on paper or a screen as you would on the day. Afterwards, spend a minute on what went well and one thing to change.
What interviewers are looking for
They are looking for how you think when the problem is new: whether you can break it down, try things, notice when something is wrong and use help to move forward. Many strong candidates do not finish every question.
They are not looking for memorised facts or a polished performance. A question you have seen before is useful only if you can explain each step, and they may simply move on to one you have not.
Curiosity counts. Asking a good question back, spotting a link to something you have studied, or saying what you would try next with more time all show the kind of student a tutor would enjoy teaching.
The online interview
Interviews are often held online now, by video call. You may be asked to write on a shared online whiteboard, to use a tablet, or to hold your paper up to the camera. Read your invitation carefully and practise with whatever it describes.
Set up the day before: check the camera, microphone and connection in the room you will use, close other programs, and keep the device plugged in. Have plain paper, a dark pen and a spare nearby. A quiet room with the light in front of you rather than behind helps them see you.
Write large and clearly, and keep talking while you write, since the interviewer cannot see over your shoulder. If you are working on paper, say what you have written and hold it up when it matters. On a shared whiteboard, practise beforehand: writing with a mouse or stylus is slower than you expect.
If the connection drops, stay calm: it happens, and interviewers are used to it. Rejoin using the same link and carry on from where you were. If you cannot get back in, use the contact details in your invitation. A dropped connection is not held against you.
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