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Some Generalizations of the Bridge and Torch Problem · arXivDesk

2608.05190Aug 2, 2026

Some Generalizations of the Bridge and Torch Problem

Pang Ern Thang, Gerard Sayson

Abstract

For the classic bridge and torch problem with crossing times $\left\{1,\ldots,n\right\}$ , we derive a closed-form expression for the optimal crossing time $T\left(n\right)$ using a recurrence relation derived from the problem's optimal substructure, thus obtaining $T\left(n\right)=\frac{n^2}{4}+3n-5+\frac{\left(-1\right)^n-1}{8}$

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For the classic bridge and torch problem with crossing times {1,…,n}, we derive a closed-form expression for the optimal crossing time T(n) using a recurrence relation derived from the problem's optimal substructure, thus obtaining T(n)=(n²/4)+3n-5+((-1)ⁿ-1/8) which holds for all n≥ 2. We generalize the problem to a bridge of capacity 3 and obtain the optimal crossing time T₃(n)=n²6+2n-(181/36)+(-1)ⁿ4-(2/9)((2nπ/3)) which holds for all n≥ 7.

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The paper addresses: Some Generalizations of the Bridge and Torch Problem.
For the classic bridge and torch problem with crossing times {1,…,n}, we derive a closed-form expression for the optimal crossing time T(n) using a recurrence relation derived from the problem's optimal substructure, thus obtaining T(n)=(n²/4)+3n-5+((-1)ⁿ-1/8) which holds for all n≥ 2.