### _h_'s blog

By _h_, history, 10 months ago,

Problem 1466I - The Riddle of the Sphinx, authored by gawry and Anadi, raised some interesting questions. The problem asks to find the maximum among $n$ secret $b$-bit $a_1 \ldots a_n$ numbers using $3n+3b$ questions of the form "is $a_i$ at most $x$"? ecnerwala described a solution using only $2n+2b-2$ questions. Here I will describe a solution using only $2n + 1.45b$ questions, using Fibonacci numbers instead of powers of two. In a sense this solution looks near-optimal, but there are better solutions when one of $n, b$ is much smaller than the other. In the other direction, a fairly simple argument shows that $n + b - 1$ questions are always needed in the worst case. I'll also sketch a proof that $n + b + 0.3 \min(n, b)$ (maybe minus some constant term) questions are needed. This shows that no solution can hope to achieve e.g $1.1 n + 1.1 b$, but still leaves a wide gap to $2n + 1.45b$.

I'll be sloppy with $+O(1)$ terms throughout, so while we are interested in the difference between $n$ and $2n$, I might forget about the difference between $n$ and $n + 10$. The ideas here arose in discussion with noncomlp. If you any ideas on how to improve the bounds in this post, please share them!

### Terminology

Throughout this post, I will be using a straightforward reformulation of the original problem. In this reformulation, the state of the game is described by a multiset $P$ of integers all at least $2$. The game ends when $P$ is empty. In one move, we may name an integer $r$. Then one of two things happen: either (one occurrence of) $\max P$ is replaced by $r$, or $P$ is replaced by $P - r := [ p - r \mid p \in P, p - r \ge 2]$. The connection with the original problem comes from the observation that instead of remembering upper / lower bounds $u_i$ / $l_i$ on $a_i$ for each $1 \le i \le n$, it suffices to remember upper bounds $u_i$ and the greatest lower bound $L$ we've seen so far. We then consider the multiset $[ u_i - L + 1 \mid u_i > L]$, and observe that asking if $a_i \stackrel{?} \le x$ is like naming $r = x - L + 1$ in the reformulation. (This relies on the observation that we can always pick the $i$ which maximises $u_i$.) Thus the number of questions needed for $n, b$ is the number of moves needed to win the game starting from the multiset of size $n$ containing just the number $2^b$.

### Fibonacci strategy

The problem hints to use powers of two, but let's instead use Fibonacci numbers. Basically this is motivated by the observation that if $\min P = F_i$ and we name $F_{i-1}$, then if we end up subtracting $F_{i-1}$, $F_i$ becomes $F_{i-2}$. Instead of starting from $P = [2^b, \ldots, 2^b]$, let's suppose we start from $P = [F_c, \ldots, F_{c+n-1}]$. I'll describe a solution which uses only $2n + c$ questions. This solves the original problem in $2n + 1.45b$ questions, where $1.45 = \log_2 \phi$, since $F_{n+1} \ge \phi^n$.

It is necessary to consider a more general class of multisets than the one above, since we might end subtracting Fibonacci numbers from much larger Fibonacci numbers. We'll assume we have some $c \ge 3$ and a string $S$ containing characters 1 and x, where no x is followed by another x. The multiset $P = P(c, S)$ is then read off as $P(c, \emptyset) = \emptyset$, $P(c, 1S) = [F_c] \cup P(c+1, S)$, $P(c, xS) = P(c+1, S) - F_c$. Thus the multiset above corresponds to the string $S$ of $n$ 1s. We claim that any such state can be done in $2n + c$ questions, where $n$ is the number of 1s in $S$.

The strategy in this class is fairly simple. We always name $F_{c-1}$. In the case where $\max P$ is replaced by $F_{c-1}$, we simply move the last 1 of $S$ to the start, and subtract one from $c$. In the case where $F_{c-1}$ is subtracted from all elements of $P$, we prepend an x to $S$ and subtract one from $c$. Now if the string looks like xx1T, then we reduce it to xT, and increase c by 2. This doesn't change $P(c, S)$ (since $F_c + F_{c+1} = F_{c+2}$), nor our monovariant $2n + c$, and after repeatedly doing this operation, no x in $S$ is followed by another x.

We should be careful to note that $c$ need never be negative, since we don't want negative Fibonacci numbers, and we also don't want our monovariant to go negative. Indeed already if $c = 2$, we can increase $c$ and remove the first one or two characters of $S$, since they give some number which is at most $1$.

Here is my implementation of this strategy: 103038134.

### Lower bounds

Here is a nice fact: if some strategy never finishes in less than $z$ moves starting from $P$, then no strategy can guarantee to finish in less than $z$ moves starting from $P$. The proof is simple. Say, starting from $P$, the first strategy names $r$, and the second strategy names $t$. If $r \le t$, consider what happens when $\max$ is replaced by $r$ or $t$: the first strategy is better off. On the other hand, if $r \ge t$, then when $r$ or $t$ is subtracted from all numbers, then the first strategy is better off.

Thus the problem of showing that no strategy always does well reduces to finding a strategy which never does well. Here is one such strategy, starting from $P = [2^b, \ldots, 2^b]$ (size $n$). If $b = 1$, then we name $r = 0$. If $b \ge 1$, then we name $2^{b-1}$. In the best case, $b$ decreases by $1$ each step until $b = 1$, at which point $n$ decreases by $1$ each step until $n = 0$. Thus $n + b - 1$ questions are needed.

The Fibonacci strategy looks like a good candidate for "strategy that never does well", since the monovariant should decrease by exactly 1 in every step. Unfortunately, it can start doing well in special cases, like when $c$ or $n$ is too small. There is a good reason for this, explained in the next section. For the rest of this section, I'll sketch a modified Fibonacci strategy which really never does too well.

Where we previously considered the alphabet with two letters 1 and x, let's now expand the alphabet to allow for arbitrary non-negative integers. Where (c, "1") corresponds to $[F_c]$, we'll let (c,"m") correspond to $[F_c, \ldots, F_c]$ (size $m$). The states we consider are given by $c\ge 3$ and strings $S$ of the form above followed by a string 0m with $m \ge 1$. Thus our initial state looks like (c, 0n). Our strategy is the same as before (instead of moving 1s from end to beginning, we replace n with (n-1) and put a 1 at the beginning), with the added reduction rule (c, x0m) -> (c, 0m) (corresponding to $F_{c+2} - F_c = F_{c+1}$). Eventually $c$ or $m$ gets small enough that we can't stay in this class of states, and our strategy might start doing well. But for the first $M := \min(c, n) - 5$ moves, we'll certainly stay in this class. Thus for the first $M$ turns, the monovariant decreases by at most 1. After these first $M$ turns, we consider the lower bound $|P| + \log_2 \min P$ from before. The most this could decrease after $M$ turns is $M \log_2 \phi$.

Taken together, this means that in the original problem with $n, b$, $n > M + 5$, $2^b > F_{M+5}$, at least $M + n + b - M \log_2 \phi > n + b + 0.3M$ questions are needed.

### Good strategies when either $n$ or $b$ is small

A silly strategy is to always name $1$. This always wins in at most $|P| + \max P - 2$ moves. So we get a lower bound on the original problem of $n + 2^b - 2$. Note that this is better than $2n$ when $b$ is very small.

At the other extreme, consider $n$ fixed. I claim that the answer is $(1 + o(1)) b$ as $b \to \infty$. The idea is to pick some positive integer parameter $D$, and consider multisets of the form $[2^k | k \in Q]$, where $\max Q - \min Q \le D$. Given such a state, I claim that we can finish in $\min Q + (D+1)|Q| + \sum Q / D$ moves. Indeed if $\max Q - \min Q < D$, then we can name $2^{\min Q-1}$, and $\min Q$ decreases by 1. If $\max Q = \min Q + D$, then we name $2^{\min Q}$. Now either $|Q|$ decreases by $1$ and $\min Q$ increases by at most $D$, or $\sum Q$ decreases by $1$.

In the original problem, we start with $\min Q = \max Q = b$, $|Q| = n$, $\sum Q = bn$. The optimal value of $D$ is about $\sqrt b$. This shows that we need at most about $2 n\sqrt b + b$ questions. This is good when $n \ll \sqrt b$.

• +195

By _h_, history, 6 years ago,

Hi! Modular inverses are used in the solutions to a lot of number theory problems, such as 622F - The Sum of the k-th Powers from the latest educational round. I want to share a one-liner (essentially) that computes modular inverse with the same complexity as the exteneded Euclidean algorithm (a and b are supposed to be positive co-prime integers, and inv(a,b) is the inverse of a modulo b, lying between 1 and b):

long long inv(long long a, long long b){
return 1<a ? b - inv(b%a,a)*b/a : 1;
}


The idea behind the code is that to compute inv(a,b), we find x such that and , and then return x/a. To find x, we can write x = yb + 1 and solve for y by recursively computing inv(b,a). (Actually inv(b%a,a))

This isn't the fastest method if you're worried about performance, and you will probably get overflow if a*b doesn't fit inside a long long. But I think it's neat.