What must be one's speed, relative to a frame S, in order....

[cosmos42]

... in order that one's clocks will lose:

(a) 1 second per day as observed from S?
(b) 1 minute per day as observed from S?

I was referencing this:( http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/tdil.html#c2 ) hyperphysics page but I still can't seem to understand what I need to do to solve for velocity.

 

[Nugatory]

In the one minute-per-day case, time dilation says that while 24*60 minutes elapse on the stationary clock, (24*60)-1 minutes elapse on the moving clock. The time dilation formula and some algebra will see you home from there.

 

[Herman Trivilino]

There are 86400 seconds in a day so on a clock that loses 1 second per day 86401 seconds will pass. Thus $\gamma$ equals $\frac{86401}{86400}$. You can then use that value in the time dilation formula you linked to and solve for $v$.

 

[PAllen]

Or, since both of your γ are close to 1, you can use the following approximate formula:

v² ≈ 2 c²(γ - 1)

 

[Ibix]

There are 86400 seconds in a day so on a clock that loses 1 second per day 86401 seconds will pass.

That's a clock gaining 1s. A slow clock will tick 86399 times in a day as measured by a clock at rest in S, which gives $\gamma =\frac {86400}{86399}$.