(Go on, solve it, you know you want to.)
Personally, I think this was the best comment:
"You do realize, we're trying to sum up Stupidity as a Mathematical Equation."
But I also like the big collection of possible approaches --- those solutions have everything!
My solution is here, hidden as blue text on blue background:
Shlemiel can walk M meters per day, and his effort can be modelled as an Arithmetic series S_n with the points S_x, S_y and S_z, therein, so we're looking for M = S_x, 2M = S_y and 3M = S_z and the values of x, y and z which give us the total distance reached on each day.
Rearranging S_n = n/2 (2p + d(n+1)) we get 2 S_n = dn^2 + dn +2pn = [(2S_n - 2pn) / d ] - n = n^2.
(where p is the initial length and d is the distance)
There is no closed form for finding n, but because n^2 is the biggest term, we can simply ignore the others and extract our values for x, y and z thus:
x = floor(sqrt(M)), y = floor(sqrt(2M)) and finally, z = floor(sqrt(3M)).
On day one he paints a = x meters, the next day he manages b = (y - a) meters, and finally, c = (z - b) meters on the third.
x = floor(sqrt(2 M/d)), y = floor(sqrt(2*2M/d)) and finally, z = floor(sqrt(2*3M/d)).
Moral of the story: beware of working with convenient values.
Update: there is of course always at least one error in every solution... and the above is no exception :)
So...
x = floor(sqrt(2 M/d)), y = floor(sqrt(2*2M/d)) and finally, z = floor(sqrt(2*3M/d)).
Moral of the story: beware of working with convenient values.
