If it takes 6 men 4 days to dig 8 holes how long does it take 1 man to dig half a hole?

The first answer is: You can't dig half a hole. As soon as you take out the first shovel-full, you have a whole hole.

But I'll play the game and show you that a mathematical solution is possible.

6 men take 4 days to do 8 jobs.
The 6 men do (8/4) = 2 jobs every day.
It takes (6/2) = 3 men to do one complete job in a day.
So each man does 1/3rd of a job each day.

We're almost there.

We know that one man does 1/3rd of a job in a day.
You want to know how long it takes for him to do 1/2 of a job.
We only need to find out how many 1/3's there are in 1/2.

(1/2) divided by (1/3) = (1/2) times (3/1) = 3/2 = 1.5

That's it.

It takes one man 1.5 days to dig 1/2 of a hole.

Check:

If that's right, then it takes him (1.5 / 0.5) = 3 days to dig a whole hole.
Six of him could dig (6 x 1) = 6 whole holes in 3 days.

In (4/3 x 3) = 4 days, the same 6 men could do 4/3 as much work ...
(4/3 x 6) = 8 whole holes instead of only 6.

Which is exactly what you described: 6 men, 4 days, 8 holes.

QED, dude.