The Apple Pear Basket Problem: A Combinatorial Exploration
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917423498657792 |
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| author | Pulikkoonattu, Rethna |
| author_facet | Pulikkoonattu, Rethna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18619 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Apple Pear Basket Problem: A Combinatorial Exploration Pulikkoonattu, Rethna General Mathematics 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. |
| title | The Apple Pear Basket Problem: A Combinatorial Exploration |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.18619 |