Problem 121: Disc Game Prize Fund

Mathematical Development

Definition 1. Let $n$ denote the number of turns. At turn $k$ ($1 \le k \le n$), the bag contains one blue disc and $k$ red discs (the original red disc, plus the $k - 1$ previously added red discs, plus the one added at the start of turn $k$), for a total of $k + 1$ discs.

Theorem 1 (Turn probabilities and independence). At turn $k$, $\Pr(\text{blue at turn } k) = \frac{1}{k+1}$ and $\Pr(\text{red at turn } k) = \frac{k}{k+1}$. The outcomes across distinct turns are mutually independent.

Proof. Consider the equivalent probabilistic model. Before turn $k$, one red disc is added to the bag and then one disc is drawn uniformly at random from the $k + 1$ discs present. Since the drawn disc is set aside and one new red disc replaces it before the next turn, the bag at each turn $k$ is deterministically composed of 1 blue and $k$ red discs, regardless of prior history. Consequently, the conditional probability of drawing blue at turn $k$ given any history equals $1/(k+1)$, and by the definition of conditional independence, the turn outcomes are mutually independent. $\square$

Theorem 2 (Winning probability). The probability that the player wins (draws at least $\lceil(n+1)/2\rceil$ blue discs in $n$ turns) is

where

and $T$ indexes the set of turns on which red is drawn.

Proof. Let $S \subseteq \{1, \ldots, n\}$ be the set of turns on which blue is drawn, and set $T = \{1,\ldots,n\} \setminus S$. By Theorem 1 and the multiplication rule for independent events,

The common denominator is $\prod_{k=1}^{n}(k+1) = 2 \cdot 3 \cdots (n+1) = (n+1)!$. The player wins when $|S| > n/2$, equivalently $|T| \le \lfloor n/2 \rfloor$. Summing the numerators $\prod_{k \in T} k$ over all qualifying $T$ yields $N$, and therefore $P(\text{win}) = N / (n+1)!$. $\square$

Lemma 1 (Elementary symmetric polynomial recurrence). Define $e(i, j)$ to be the $j$-th elementary symmetric polynomial of $\{1, 2, \ldots, i\}$, i.e., the sum of products of all $j$-element subsets. Then

with boundary conditions $e(0, 0) = 1$ and $e(i, j) = 0$ for $j < 0$ or $j > i$. Moreover,

Proof. Partition the $j$-element subsets of $\{1, \ldots, i\}$ by membership of $i$:

• Subsets not containing $i$: these are $j$-element subsets of $\{1, \ldots, i-1\}$, contributing $e(i-1, j)$.
• Subsets containing $i$: formed by selecting a $(j-1)$-element subset of $\{1, \ldots, i-1\}$ and multiplying by $i$, contributing $i \cdot e(i-1, j-1)$.

Since each subset $T$ of $\{1, \ldots, n\}$ with $|T| = j$ contributes $\prod_{k \in T} k$ to $e(n, j)$, summing $e(n, j)$ for $0 \le j \le \lfloor n/2 \rfloor$ yields $N$. $\square$

Theorem 3 (Maximum prize fund). The maximum prize fund is $\lfloor (n+1)! / N \rfloor$.

Proof. The banker charges $1 per game. For a prize of$M$dollars, the expected payout is$M \cdot P(\text{win}) = MN/(n+1)!$. Non-negative expected profit requires$MN/(n+1)! \le 1$, hence$M \le (n+1)!/N$. The maximum integer prize is therefore$\lfloor (n+1)!/N \rfloor$.$\square$

Editorial

At turn k, P(blue) = 1/(k+1), P(red) = k/(k+1). Player wins with >= 8 blue out of 15 turns. Prize = floor((n+1)! / N) where N sums elementary symmetric polynomials.

Pseudocode

    dp[0..floor(n/2)] initialized to 0
    dp[0] = 1

    For i from 1 to n:
        For j from min(i, floor(n/2)) down to 1:
            dp[j] = dp[j] + i * dp[j-1]

    N = sum(dp[0..floor(n/2)])
    D = (n+1)!
    Return floor(D / N)

Complexity Analysis

• Time: The double loop performs $\sum_{i=1}^{n} \min(i, \lfloor n/2 \rfloor)$ iterations, which is $O(n \cdot \lfloor n/2 \rfloor) = O(n^2)$.
• Space: $O(n)$ for the one-dimensional DP array, updated in-place by processing the inner index in decreasing order.

Answer

#include <bits/stdc++.h>
using namespace std;

int main() {
    const int n = 15;
    const int half = n / 2; // 7

    // dp[j] = j-th elementary symmetric polynomial of {1,...,i}
    vector<long long> dp(half + 1, 0);
    dp[0] = 1;

    for (int i = 1; i <= n; i++)
        for (int j = min(i, half); j >= 1; j--)
            dp[j] += (long long)i * dp[j - 1];

    long long N = 0;
    for (int j = 0; j <= half; j++)
        N += dp[j];

    // Compute (n+1)! = 16!
    long long fact = 1;
    for (int i = 1; i <= n + 1; i++)
        fact *= i;

    cout << fact / N << endl;
    return 0;
}

"""
Problem 121: Disc Game Prize Fund

At turn k, P(blue) = 1/(k+1), P(red) = k/(k+1).
Player wins with >= 8 blue out of 15 turns.
Prize = floor((n+1)! / N) where N sums elementary symmetric polynomials.
"""

from math import factorial

def solve():
    n = 15
    half = n // 2  # 7

    # dp[j] = e(i, j) = j-th elementary symmetric polynomial of {1,...,i}
    dp = [0] * (half + 1)
    dp[0] = 1

    for i in range(1, n + 1):
        for j in range(min(i, half), 0, -1):
            dp[j] += i * dp[j - 1]

    N = sum(dp)
    D = factorial(n + 1)
    print(D // N)

solve()