A puzzle at the end of a 3Blue1Brown video asks the following question (paraphrased):

From a group of 20 people, you get to send one person to participate in a tug-of-war tournament. You don't care too much who you send, as long as you don't send the weakest person. Each person has a different strength, but you don't know what it is. You get 10 tug-of-war matches of 10 vs. 10 among the group of 20 to determine who to send. How do you make sure you don't send the weakest person?

I won't spoil the solution to the puzzle, but it will likely come as no surprise that the (or at least, my) solution works for any group of $2n$ people, where you get $n$ matches of $n$ vs. $n$ people.

Can you do any better, for sufficiently large $n$?

Is there an $n$ such that you can find a non-weakest person among a group of $2n$ people, using only $n - 1$ tug-of-war matches of $n$ vs. $n$?

And, if this question has an obvious answer I did not think of, can we even characterize the behaviour?

Let $L(k)$ be the least number of matches required to find a non-weakest person among $2k$ people. What is the asymptotic behaviour of $L$?

  • $\begingroup$ Can team composition (i.e. how to divide into two teams) for round $k$ depend on previous results? If so, I think I have a $\Theta(\log_2 n)$ solution. In fact I think it is $1 + \log_2 n$. $\endgroup$ – antkam Mar 27 at 17:33
  • $\begingroup$ @antkam, I definitely think it can, and intended my question this way, although my solution to the original puzzle does not require it -- so perhaps it makes for another interesting puzzle. $\endgroup$ – Mees de Vries Mar 27 at 17:36
  • $\begingroup$ (1) Curious about your spoiler: Is it equivalent to sitting the people in a circle, the red team being half the circle, and in each round shifting the red team by $1$ person clockwise? :) (2) I agree that if team membership in all rounds must be fixed before any matches, the problem is much tougher. Off the top of my head, I can't think of anything that works in $<n$ rounds in that model, but it might be possible...? $\endgroup$ – antkam Mar 27 at 20:49
  • $\begingroup$ @antkam, that was indeed my solution. $\endgroup$ – Mees de Vries Mar 28 at 16:09

This answer assumes we can decide how to divide into two teams for round $k$ based on previous rounds' results.

Notations: Let $X, Y$ be sets of people. I will abuse notation a bit and write $X+Y$ for their combined strength. Also, suppose $X$ consists of $n$ people, then a match can be held vs the other half, which I denote as the complement $X^c$. Further suppose $X$ wins this match. I will write $X > X^c$, or $X > 1/2$ meaning $X$ has more than half the total strength. In other words, $X$ will stand for the set if set-theoretic operations are performed (unions, etc), but $X$ will actually stand for the total strength of the people $\in X$ if arithmetic operations are performed.

Assume for now $n = 2^b$ for integer $b$. Here is a solution that requires $1+b$ rounds.

First partition the $2n$ people into $4$ equal sets $A, B, C, D$ each of size $n/2$.

Round $0$: $A \cup B$ vs $C \cup D$.

Round $1$: $A \cup C$ vs $B \cup D$.

One of the groups has to win twice, and wolog assume it is $A$. Then we have $A + B > 1/2 > B + D$.

Define a triplet of subsets $(X,Y,Z)$ to be an $m$-triplet if (1) the $3$ sets are disjoint, (2) they have sizes $(m, m, n-m)$ respectively, and (3) $X+ Z > 1/2 > Z + Y$. So at the end of Round $1, (A,D,B)$ is an ${n\over 2}$-triplet.

The recurrence step will be of this format: Suppose we have an $m$-triplet $(X, Y, Z)$. Divide $X$ into $X_1, X_2$ each of size $m/2$, and $Y$ into $Y_1, Y_2$ each of size $m/2$. Now match up $W = X_1 \cup Y_2 \cup Z$ vs $W^c$.

  • Case 1: If $W$ wins, then $W = X_1 + Y_2 + Z > 1/2$. But we know $1/2 > Y + Z = Y_1 + Y_2 + Z$. So we now have a new ${m\over 2}$-triplet $(X_1, Y_1, Z \cup Y_2)$.

  • Case 2: If $W$ loses, then $W = X_1 + Y_2 + Z < 1/2$. But we know $X + Z = X_1 + X_2 + Z > 1/2$. So we now have a new ${m\over 2}$-triplet $(X_2, Y_2, Z \cup X_1)$.

So with one match, we reduced an $m$-triplet to an ${m\over 2}$-triplet.

Starting with $n = 2^b$ (recall there were $2n$ people total), after the first $2$ matches we have a $2^{b-1}$-triplet, and the exponent drops by $1$ every additional match. After $b-1$ more matches we have a $2^0$-triplet, i.e. a $1$-triplet $(F,G,H)$ where $F+H > 1/2 > G+H$. But this implies $F > G$, and since each set contains only one person, we know the person in $F$ is not the weakest since the person in $G$ is weaker.

Total number of matches $= b + 1 = 1 + \log_2 n$.

If $n$ is not a power of $2$, the procedure is a bit messier, but we ultimately still rely on the same recurrence. If $m$ is odd, we divide $X$ and $Y$ as evenly as possible, where the sizes of $X_1$ and $Y_1$ are $\lceil {m\over 2} \rceil$. Then in case 1 we get a new $\lceil {m\over 2} \rceil$-triplet whereas in case 2 we get a new $\lfloor {m \over 2} \rfloor$-triplet. Clearly the total number of matches $= 1 + \lceil \log_2 n \rceil$.

  • $\begingroup$ nice! And in a way you could see this as a generalization of my solution. If $n = ab$, with $a, b > 1$ integers, you can first do the sliding window around the table in $b$ steps of $a$ people. Once you know where the jump is you need only $b - 1$ sliding window steps to get the exact person. Two more observations: you can iterate this, so if $b = a'b'$ you can reduce steps further; and $ab$ only needs to be an upper bound of $n$, not an exact product. Then your solution is this method applied to the product decomposition $n \leq 2^{\lceil \log_2(n) \rceil}$. $\endgroup$ – Mees de Vries Mar 28 at 16:12
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    $\begingroup$ @MeesdeVries - that's exactly the picture I had in mind. First I found your "circle, shift-by-1" solution. Then I convinced myself shift-by-2 works. Then it's pretty obvious to try either shift-by-$\sqrt{n}$ or binary-search. I tried the latter and it worked. For the written Answer above, I just decided the set-based presentation is easier than the circle-based presentation, but in fact I think we can line up all the $X$s and $Y$s and $Z$s at all levels of the recursion in a line, and show it is "embeddable" into a circle, so to speak. $\endgroup$ – antkam Mar 28 at 17:21

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