Problem 4c of Ps2 asked to construct a rotation of π/4 about the axis (x̂ + ŷ)/√2 in terms of H's and T's.

The solution I had in mind is as at left, using T to first rotate the (x̂+ ŷ)/√2 axis to the x̂ direction, then use HTH to rotate by π/4 about that new x̂ direction, then use T⁷=T^-1 to restore that x̂-axis to the original (x̂+ ŷ)/√2 axis; with the end result a rotation by π/4 about that axis. This required a total of eleven T's and H's.

But when someone mentioned it might be possible with less than eleven, I did a quick exhaustive search for all possibilities of up to eleven T's and H's (only 4095 total including the identity, so simple to run), and turned up the additional possibility THTTHTTTHT, with just ten operations. This was difficult to interpret, until rewriting as (THT)²T(THT)= (THT)^-1 T (THT), including recognizing that THT is a rotation by 2π/3 about the axis (2x̂+ ŷ)/√3, which takes the (x̂+ ŷ)/√2-axis up to the ẑ-axis. This was demo'd interactively at the beginning of lec11 in class, showing what axes looked like from various viewpoints.
The interpretation of the visualization at right is first use THT to rotate the (x̂+ ŷ)/√2-axis to the ẑ-axis, use T to rotate by π/4 about that ẑ-axis, then use (THT)²=(THT)^-1 to restore that ẑ-axis back to the original (x̂+ ŷ)/√2 direction.

[Note: strictly speaking given the convention that rotations are clockwise looking down on the axis, THT should be called a rotation by 4π/3 about the (2x̂+ ŷ)/√3-axis, but that doesn't affect the end results of the rotations]