Most of my experiences with solving mathematical problems have been challenging at first -- but often, after I explore and collect my thoughts, a pattern emerges. The notion of "difficult at first" is vividly expressed in the following poem (found in the anthology Against Infinity (edited by Ernest Robson and Jet Wimp, now available at various used-book sites).
Geometry Test by Larry Rubin
Thirty minutes, we had, to prove the theorem.
For twenty I sat staring at circles,
My inner angles frozen
When nothing came out equal.
The bisectors I drew were tilted wrong
While fear of the circular face of time
Stiffened my blood like clock-hands
Tracing arcs I never knew existed.
Suddenly that curve stretched perpendicular --
Longer that my longest transverse line --
Reaching beyond the limits of the page;
And the tallest segments of the intersected cone
Slit the seal of infinity.
My mind was washed like windshields after rain
And circles glided smoothly into place,
The arcs connecting in their shrunken frames,
I left that room, all theorems proved.
For twenty I sat staring at circles,
My inner angles frozen
When nothing came out equal.
The bisectors I drew were tilted wrong
While fear of the circular face of time
Stiffened my blood like clock-hands
Tracing arcs I never knew existed.
Suddenly that curve stretched perpendicular --
Longer that my longest transverse line --
Reaching beyond the limits of the page;
And the tallest segments of the intersected cone
Slit the seal of infinity.
My mind was washed like windshields after rain
And circles glided smoothly into place,
The arcs connecting in their shrunken frames,
I left that room, all theorems proved.
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