Investment Banking Job Interview Question: Counting Up

Analysing this simple number-based strategy game isn’t too difficult outside the stressful context of an investment banking job interview, so put yourself under some time pressure to come up with the answer…

Question:
We shall play a competitive game, taking turns to call out integers. The first person to call out “50” wins. The rules are as follows:

  1. The player who starts must call out an integer between one and 10, inclusive
  2. A new number called out must exceed the most recent number called by at least one and by no more than 10. For example, if the first player calls out “nine,” then the range of valid numbers for the opponent is 10 to 19, inclusive.

Do you want to go first, and if so, what is your strategy?

Answer:
You get the answer by working backwards. If I am your opponent, and I am able to call out “39,” then you cannot reach 50, but I can after you say whatever you say. So, my goal is to call out “39.” However, if I am able to call out “28,” then you cannot get to 39, but I can after you say whatever you say. So, my goal is to call out “28.” To get to 28, I need only to be able to call out “17,” and to do this, I need only to be able to call out “6.”

So, my strategy, as your opponent, is to get onto the series 6, 17, 28, 39, 50 at whichever point I can. If you get to go first, you should call out “6.” As long as you know the winning numbers and stick to them, you cannot lose. If you start with anything other than 6, I cannot lose.

Special thanks for this interview question to Timothy Crack, author of Heard on the Street: Quantitative Questions from Wall Street Job Interviews.

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