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arXiv · 2604.18619

The Apple Pear Basket Problem: A Combinatorial Exploration

Rethna Pulikkoonattu
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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.

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Rethna Pulikkoonattu. 2026-04-17. The Apple Pear Basket Problem: A Combinatorial Exploration. https://arxiv.org/abs/2604.18619

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