To solve the problem for the case where ( N = 1 ), we analyze the conditions under which you can guarantee victory.

Key Observations for ( N = 1 ):

• There are 2 people (you and one friend). The total check is ( T ), split equally, so each pays ( P = \frac{T}{2} ).
• You want to eat more than you pay (( x > P )) and not be exposed (your eaten amount ( x ) is at most ( B ) more ...

I’ll solve the two–player case (N = 1) completely. This is the version where you and one friend alternate ordering 1..A dollars at a time until the running total hits exactly T, and you split the check 50–50. You want to guarantee, no matter how your friend orders, that either:

• you eat more than T/2 but are not “exposed” (i.e., your total does not exceed your friend’s by more than B), or
• if th...

Below I treat the case N = 1 (there are only two diners, you and one friend) completely.
I prove that with the simple “compliment–to–(A+1)” strategy you either

• always eat strictly more than the half you must pay for, while
never exceeding the next biggest eater by more than B, or

• you can be forced to fail.

The result is given as a pair of easy-to-check inequalities that involve only T...

Below is a complete solution to the “one‐friend’’ case, N = 1. In that case you and exactly one opponent alternately order dishes costing 1…A dollars until the running total reaches T, and you win if at the end either

• you have eaten strictly more than half the total (e₀ > T/2) but not more than B dollars over your friend
i.e. 0 < D ≡ e₀–e₁ ≤ B,

• or (if you cannot achieve the ab...

Looking at this problem, I need to analyze when you can guarantee victory as the first player in this sequential ordering game.

Let me start by understanding the victory conditions:

1. Primary goal: Eat more than B of the next highest eater
2. Secondary goal: If (1) fails, see someone else exposed (eating more than $B above the next highest)

Analysis for...

This is a fascinating problem that combines elements of combinatorial game theory with strategic optimization. Let's break it down, starting with the N=1 subproblem as requested, and then generalizing.

Core Concepts and Definitions

First, let's formalize the victory and loss conditions for you (the first player).

• Players: You (Player 0) and N friends (Players 1 to N). Total `P = N+1...

<think> I need to solve this problem about a glutton trying to outsmart his friends when splitting a check. The goal is to either eat more than his fair share without being exposed or have someone else exposed as a glutton. Being exposed means eating more than $B extra compared to the next most expensive eater.

First, I should understand the setup. There are N friends plus me, so total of N+1 peo...

A Glutton and a Split Check - Mathematical Analysis

This problem is a fascinating strategic puzzle about how to maximize your share of a restaurant meal without being exposed as a glutton. Let me analyze it systematically.

Understanding the Problem

You and N friends agree to split a 1 and $A when the waiter comes to them. Your go...

Below is a complete solution of the two–person sub-problem (N = 1).
(The general case N ≥ 2 is considerably messier; the method that follows is the standard tool one would start from, but the bookkeeping with N > 1 soon becomes prohibitive, so I do not claim a finished answer in that case.)

Notation and conventions

• A ( ≥ 1) – the menu limit: on every turn a player must order an integer n...