In Season 2 of Squid Game, contestants play a game of “Mingle” (spoiler alert!). For each round of this game, the contestants all wait in a common area until a number is called. They must then split up into groups of that number. Any contestants not in a group of the correct number after 30 seconds … well, let’s just say bad things happen to them. For now, we’ll refer to contestants who make it through a round in a group of the correct size as having “survived” the round.
Suppose there are initially N contestants.
In the first round, contestants must form groups of 1. Okay, that’s not too hard; everyone survives. In the second round, contestants must form groups of 2. This continues (with groups of k in round k) until the 38th round, when contestants must form groups of 38.
What is the smallest number N such that there is at least one person who survives all 38 rounds?
If at least one person survives all 38 rounds, at least 38 people survive all 38 rounds.
The minimum number of people in the previous round is the smallest multiple of 37 that is greater than 38. Given there are $g_k$ groups of contestants surviving round $k$, the minimum number of groups of contestants that survive round $k-1$ is :
$$g_{k-1} = \bigg\lceil \frac{g_k \times k}{k-1}\bigg\rceil$$I created code that did the following :
454 is the smallest number of starting contestants in order to have reach round 38.
There are N contestants about to play Mingle, where N is less than 500. This time, in each round, the number for group size can be anywhere from 1 to 10, with each number equally likely to be called. Also, the numbers called in the rounds are all independent of each other. It appears this version of Mingle will continue until there are no more survivors.
Before the game starts, one of the contestants, Young-il, does some quick computation. He says to his companion, Gi-hun: “Oh no, it’s too bad we only have N people.” (Of course, Young-il doesn’t actually say “N,” but rather the number that N represents.)
Young-il continues: “If we had started with N+1 people instead, I’d expect the game to last about 10 additional rounds, on average, than I’d expect from starting with N people.”
What is the value of N?
The expected number of Rounds for $N$ contestants playing this version of Mingle can be represented by:
$$EV_N = 1 + \sum\limits_{g = 1}^{10} \frac{1}{10}EV_{M(N,g)}$$
where $g$ is the group size called and $M(N,g)$ is the greatest multiple of $g$ less than $N$. Note that if $g$ is a factor of $N$:
$$EV_M(N,g) = EV_N$$
These rounds have no one eliminated, and thus prolong the game.
I created code that did the following :
I noticed that $EV_p = EV_{p-1}$ for prime $p$. This is because the largest multiples of $2$ through $10$, less than $p$, are the same as that of $p-1$.
I hypothesized that the largest increase in additional rounds for $+1$ starting contestants occurs with contestant values of $p+1$, particularly $p+1$ that are somewhat overlapping with highly composite and super abundant numbers.
I looked at starting values of $1$ to $1000$. If there are at least $3$ additional rounds between round $r$ and $r+1$, it is in the table.
| $r$ | $r+1$ | Round Difference | $r$ is prime? | $r+1$ is superabundant? | $r+1$ is highly composite? |
|---|---|---|---|---|---|
| $71$ | $72$ | $3$ | Yes | No | No |
| $119$ | $120$ | $5$ | No | Yes | Yes |
| $143$ | $144$ | $3$ | No | No | No |
| $179$ | $180$ | $4$ | Yes | Yes | Yes |
| $209$ | $210$ | $3$ | No | No | No |
| $239$ | $240$ | $5$ | Yes | Yes | Yes |
| $251$ | $252$ | $3$ | Yes | No | No |
| $279$ | $280$ | $3$ | No | No | No |
| $359$ | $360$ | $10$ | Yes | Yes | Yes |
| $419$ | $420$ | $4$ | Yes | No | No |
| $449$ | $450$ | $3$ | Yes | No | No | $479$ | $480$ | $4$ | Yes | No | No |
| $503$ | $504$ | $5$ | Yes | No | No |
| $539$ | $540$ | $5$ | No | No | No |
| $559$ | $560$ | $3$ | No | No | No |
| $599$ | $600$ | $4$ | Yes | No | No |
| $629$ | $630$ | $4$ | No | No | No |
| $719$ | $720$ | $9$ | Yes | Yes | Yes |
| $791$ | $792$ | $3$ | No | No | No |
| $809$ | $810$ | $3$ | Yes | No | No |
| $839$ | $840$ | $10$ | Yes | Yes | Yes |
| $899$ | $900$ | $5$ | No | No | No |
| $959$ | $960$ | $4$ | No | No | No |
There seems to be something to it...