The Apple Pear Basket Problem: A Combinatorial Exploration

TL;DR

This paper explores a combinatorial puzzle involving distributing apples and pears into baskets with specific constraints, revealing a mathematical relationship tied to divisibility and number classification.

Contribution

It establishes a formula for the maximum number of baskets based on divisors of N and classifies integers by packing efficiency, supported by computational data.

Findings

Maximum number of baskets equals the largest divisor of N not exceeding (1 + sqrt(1+8N))/2.

For N=60, the maximum is 10 baskets.

Computational results up to N=1,000,000 confirm the asymptotic growth rate of sqrt(2N).

Abstract

We investigate a combinatorial puzzle in which $N$ apples and $N$ pears are distributed among baskets subject to two constraints: every basket must contain the same number of apples, and every basket must contain a distinct number of pears. We prove that the maximum number of baskets is the largest divisor of $N$ not exceeding $(1 + \sqrt{1+8N})/2$. For the original puzzle with $N = 60$, this yields 10 baskets. The solution reveals a rich interplay between divisibility and combinatorics, leading to a natural classification of integers into perfect values, primes, and highly composite numbers according to their basket-packing efficiency. Computational results for $N$ up to one million confirm the asymptotic growth rate of $\sqrt{2N}$, and a complete tabulation for $N = 1$ to 100 is included.