Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form.  He could just "see" the answer.  There are many mathematicians with extraordinary powers of mental visualization, even if they may not have been superb at symbol manipulation.

I'm interested in bona fide mathematical problems where one can visualize the answer, whether or not one later resorts to algebraic manipulation or proof.  Certainly some mathematical disciplines are more amenable to the techniques of visualization (geometry, topology, ...) than others (number theory...).  I'm not interested in the myriad problems posed from *other* fields, such as physics and chemistry, where the core problem is itself geometric, spatial or visual.

I'm interested in collecting problems in different branches of mathematics that are amenable to "purely visual" solution (or at least visual reasoning), even if formal proof may come later.  Here are a few that come immediately to mind.  I'll add such problems to the other subfields as I come across them.  Of course the scale of these problems cannot be excessive (e.g., you cannot present a knot diagram with 1000 crossings and ask a student to visualize whether it is the trivial knot.)

My overall goal is to help students develop their mathematical visualization skills.

**Knot theory**

 - Determining if a knot diagram describes the trivial knot 
 - Determining if two knots are related by a mirror symmetry 
 - Determining the number of components from a link diagram 
 - Assigning colors to segments in a tri-colorable knot diagram
 - Determining which crossing can be changed to make a knot trivial
 - Determining if a two-component link is separable
 - Determining if two knot diagrams describe the same knot, as in this case:

[![enter image description here][1]][1]

**Graph theory**

 - Determining if a graph (represented with arbitrary vertex locations) is in fact planar, e.g., this one:

[![enter image description here][2]][2]

**Discrete group theory**

 - Determining if a particular symmetry of group operation transforms one given geometric figure to another

**Geometry**

 - Determining the general shape of the intersection of two solids or surfaces (e.g., a plane and a cone)
 - Determining the three-dimensional form (e.g., of a polyhedron) from a planar map of its faces e.g.:  Which of these figures can be folded into a cube?

[![enter image description here][3]][3]

**Number theory**

**Calculus**

**Linear algebra**

**Differential equations**

**Dynamical systems**

**Game theory**

**Probability and statistics**

**Real analysis**

**Algebraic geometry**

**Complex analysis**

**Combinatorics**

**Fiber bundles and cobordism**

**Category theory**


  [1]: https://i.sstatic.net/IgExI.png
  [2]: https://i.sstatic.net/8qP1S.png
  [3]: https://i.sstatic.net/s1tOf.png