The Traitorous Generals

This Riddler problem is a logic puzzle about liars and truth-tellers.

You are the emperor of Byzantium (lucky you!) and you have N generals working for you. You know that more than half of your generals are loyal, and the rest are traitors. You can ask any general about the loyalty of any other general: If the general you ask is loyal, he will answer correctly, but if he is a traitor he can answer however he likes. Your goal is to find one general you are absolutely certain is loyal while asking the fewest possible questions.

What is the minimum number of questions (in terms of N) that will guarantee a solution, and what strategy produces it?

There is a problem in cryptography known as the Byzantine Generals Problem, which has to do with achieving consensus in the presence of traitors that can sabotage communications. The Riddler problem above also involves liars and truth-tellers, but it’s a very different problem.

The following is adapted from a comment by Guy Moore. A similar solution that obtains the same final result was also found by Dmytro Taranovsky. Thank you both for your insights!

[Show Solution]

11 thoughts on “The Traitorous Generals”

  1. On your way towards asking the 2k generals, as you say, the worst case is that all traitors tell the truth. But then you can stop after the k+1st answer, no?

  2. This is an interesting problem, but your solution is not optimal. Assuming that questions can depend on previous answers, I got a strategy that uses at most
    N-1-f(N-1) questions where f(k) is the number of ones in the binary representation of k. I do not know whether it is optimal.

    Interesting Variations:
    * You can use randomness in your strategy. The worst case appears to be when the number of disloyal generals is maximum and disloyal generals answer incorrectly; the expected number of questions is then 3/4*N-o(N).
    * Questions cannot depend on previous answers. Your strategy has this property, but it is not optimal either. If k=1, it is enough to ask one question: If A says that B is loyal, then B is loyal, and otherwise C is loyal. If there is a constant c>2 such that N/k>c (where k is number of traitors), then I think O(N) questions suffice (with a majority of possible choices of the set of questions guaranteed to work), but I do not have a general answer.

    1. Here is my strategy (do not read if you want to figure it out on your own):
      For each general, assign a number, initially 1.
      While the highest number is less than half of the sum of all numbers:
      Pick any two generals with the same number for the highest number that such a pair exists, and ask the first one whether the second one is loyal
      – if no, remove both generals (and their numbers)
      – if yes, remove the first general and double the second general’s number
      Pick a general with the highest number.

      Also, correction: For the problem variation that allows randomness, the expected value should have been 2/3*N-o(N).

    2. I agree with Dmytro Taranovsky. The optimal solution requires N-1-f(N-1) questions.
      The key is to efficiently reduce the number of generals while maintaining the property that at more than half of the generals who remain are loyal. A step in this procedure is, for N odd, to break the generals into floor(N/2) pairs plus a leftover. (If N is even, ignore one general and solve the problem with the remaining N-1 generals.) For each pair of generals, ask one if the other is loyal. If the answer is “no,” then at least one is disloyal. Discard the pair; among all remaining generals we still have a majority of loyal generals. Now build a set using the generals whose colleague claimed that they were loyal, plus the leftover general if the number of claimed-loyal generals is even. This set has at most 1+2 floor((N-1)/4) generals and the majority are loyal. (If a general is claimed loyal but is really a traitor, then both generals in that pair were traitors. At most half of the pairs can have this property, and if exactly half of pairs do have this property then the leftover general is loyal and is included.)
      Therefore, using floor((N-1)/2) questions, we have lowered the number of generals from N to at most [1+2 floor((N-1)/4)]. This procedure will indeed lower the number of generals to 1 (who is then loyal) using at most N-1-f(N-1) questions.
      I don’t see how to use randomness in the strategy, though. Maybe you could clarify?

      1. Thank you both for your insights! Guy and Dmytro, I removed my solution which was quite suboptimal and I posted a write-up adapted from the comments above. Thanks again!

      2. It’s difficult to make sense of the layers of complexity imbued in portions of the response, but clearly it has an audience of considerate choir members.

        Conventionally the responses appear both constrained and simple:

        N of Questions = the total (N)umber of Generals when the total N of generals is odd, accounting for more than 2.

        N of Questions = total (N)umber of generals minus one when the total N of generals is even, accounting for more than 2.

        N of Questions= Zero when the total (N)umber of generals is 2 or less.

        There is another prospect, and it is far more interesting. The unconventional solution is constant irrespective of the total (N)umber of generals:
        N of Questions is ONE, FOLLOWING ONE COMMAND. There were no limitations placed on commamds. Any rate, it can go as follows:

        Irrespective of the number of N, only one command and one question need be posed one time in a whole group lineup of generals:

        (COMMAND)
        Form a complete cohort/grouping of generals that are loyal to me by holding hands only with loyal generals, closing your circle only when all loyal generals are holding hands, and do not answer me until you have completed this task with absolute accuracy!

        (QUESTION)
        Are there any disloyal generals among you? / Has every loyal general identified every other loyal general?

        Now then, let us usher in the world of Ultimate Prosperity!

        With that, since all of the loyal generals can only speak correctly, they would only answer when they have achieved the accurate accounting of all loyal generals, which incidently they would all know once asked, if not earlier.

        The result will be that the grouping of identified loyal generals will represent a quantity greater than half the total number of generals.

        This beneficent leader would prefer to save both his pipes and time, alotting each for useful questions and instructions given to the most competent of the loyal generals so that we can Harmonize the World with Plans that will enable it to experience peaceful abundance. But whatever the leader’s ambitions, with the unconventional solution, they will be freed to identify their preferred traits among the full spectrum of loyal generals available. To be sure, some of the so-called leaders will waste plenty of talent due to their preference for irrelevant traits and inherent distrust of people with qualities they somehow find personally objectionable, despite the fact that such traits do not otherwise disfavorably impact a general’s competence, capacity nor performance.

        N=ONE!

  3. As I am of considerably low mathematical maturity, I was wondering whether anybody here can go into the details of the solutions including steps that might be to you relatively trivial. I worked on this problem for quiet a few hours so I want to understand how it is solved starting with the statement about the floor being 1+2floor

  4. I am not convinced about the majority argument. Say there are 5 generals. We group them as LT LL and T. In the first pair both of them will be eliminated. In the second pair only first general will be eliminated. We will be left with LT and majority is lost for loyal. The problem persists in all case when traitor is set aside. Among the pairs all are LT and one pair is LL. Then all LTs will be eliminated and we will be left with one L and one T.

    1. Thanks for the comment — I fixed the solution and it should be correct now. Basically, you only keep the general that was set aside if there is an even number of generals remaining.

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