Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics questions you've found that can be better solved with a concept from physics – or at least where the application of physics is interesting and perhaps illuminating.
Examples
One of these questions is on minimizing the time taken for a lifeguard to go out to a stationary distressed swimmer. In the scenario, the lifeguard runs faster than he swims in the water, and as such the straight line is not the fastest way for the lifeguard to reach the swimmer. Students will normally use calculus to solve this problem, and the answer can be obtained after some work – however, a much more convenient (and intuitive) way is to borrow from the idea of refractive indices in geometric optics. We recast the situation by replacing the beach and the sea with two materials with different refractive indices, choosing the appropriate refractive indices proportionate to the ratio between the lifeguard's velocities while running and swimming. The problem is then reduced to finding a beam of light that passes through both the swimmer and the lifeguard's position. (For a more complete explanation, you can visit this site: http://findingmoonshine.blogspot.sg/2012_05_01_archive.html)
Another of these questions requires one to prove that, in an acute-angled triangle, the angle subtended by any side of the triangle at the Toricelli point is 120°. Again, instead of using trigonometry, one can use the concept of hanging equal weights from a (frictionless) string at each of the vertices of the triangle, and then tying each of the three strings together at one knot placed on the surface of the triangle. The equilibrium position of the knot is the Toricelli point, and one can then complete the proof by considering forces acting on the knot.
Looking forward to hearing from you!
Best Answer
Personally speaking, I very rarely use Physics directly to solve Mathematics problems (apart from your example, which I used before). I do find that with very (mathematically) abstract PDE's/dynamical systems, a basic understanding of physics greatly enhances my intuition on the subject. So it helps me indirectly, but in a powerful way.
To give you a better example though, consider an analytic (read: differentiable) function $f(z)$ from $\mathbb{C}$ to $\mathbb{C}$. Many first time students struggle to visualise what this function represents, and in particular what the integral of the function represents. What does $\int\! f(z) \, \mathrm{d}z$ mean?
Physically there is a beautiful answer. Define $\overline{f(z)}$ to be the Polya vector field of $f$, where $f$ is analytic. Then in the complex plane, $\overline{f}$ is a sourceless, irrotational vector field. Astounding! What's more: $$\int_C f(z) dz=\mathcal{W}[\overline{f},C]+i\mathcal{F}[\overline{f},C]$$ where $C$ is any arbitrary curve in the plane, $\mathcal{W}$ denotes the work done along $C$ and $\mathcal{F}$ denotes the flux passing through $C$. So now complex integrals should be second nature to any physicist, particularly those who like Electromagnetism. See here for more: http://demonstrations.wolfram.com/PolyaVectorFieldsAndComplexIntegrationAlongClosedCurves/
As an example, Cauchy's Theorem states that $$\int_C f(z) dz=2\pi i \sum\limits_i Res[f(z),z_{i}]$$ where $C$ is a simple closed contour. This is a nice result, but isn't very intuitive... at least initially. Recall that $\frac{1}{z}=\frac{\bar{z}}{\lvert z \rvert ^2}$ so the Polya vector field of $f(z)=\frac{1}{z}$ is $\frac{z}{\lvert z \rvert ^2}$ which corresponds to a point charge at the origin. Now we know by Gauss' law that the flux through any closed contour should be equal to the some of the charge inside, proving Cauchy's theorem! For more on this overlap, see Tristan Needham's Visual Complex Analysis.