The Apple Pear Basket Problem: A Combinatorial Exploration

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.