I hadn't read that line about math.floor, because it is preceded by "Note for JavaScript, Coffescript, Typescript...." Didn't know that there is information needed for Python solution also.

The progression that @Pigankle posted uses floating-point numbers, and does not round down at each step. The problem statement doesn't ever explicitly say that it expects you to round your guesses to integer at each step, but the final line (suggesting the use of Math.floor "for each integer approximation") is trying to suggest that. However, the problem statement does need to be a bit clearer about saying "Your guesses should be integers at each step".

I am seeing the same result as Arcuo. Hero's Method converges in 29 steps for the input parameters given in that test. I printed out each step and counted them. I double checked this with excel. If the five of us who are getting 29 were instead writing code that converged too slowly, I would think we are doing something wrong. As is, we are converging faster than others, and I cannot see how we could be incorrect.

Iteration	 n 	              x 	                     newx	                 error after this iteration 
 1	 835,871,232,077,058 	 1.0                    	 417,935,616,038,529.0 	 417,935,616,038,528.0 
 2	 835,871,232,077,058 	 417,935,616,038,529.0 	 208,967,808,019,266.0 	 208,967,808,019,264.0 
 3	 835,871,232,077,058 	 208,967,808,019,266.0 	 104,483,904,009,635.0 	 104,483,904,009,631.0 
 4	 835,871,232,077,058 	 104,483,904,009,635.0 	 52,241,952,004,821.4 	 52,241,952,004,813.4 
 5	 835,871,232,077,058 	 52,241,952,004,821.4 	 26,120,976,002,418.7 	 26,120,976,002,402.7 
 6	 835,871,232,077,058 	 26,120,976,002,418.7 	 13,060,488,001,225.4 	 13,060,488,001,193.4 
 7	 835,871,232,077,058 	 13,060,488,001,225.4 	 6,530,244,000,644.7 	 6,530,244,000,580.7 a
 8	 835,871,232,077,058 	 6,530,244,000,644.7 	 3,265,122,000,386.3 	 3,265,122,000,258.3 
 9	 835,871,232,077,058 	 3,265,122,000,386.3 	 1,632,561,000,321.2 	 1,632,561,000,065.2 
10	 835,871,232,077,058 	 1,632,561,000,321.2 	 816,280,500,416.6 	 816,280,499,904.6 
11	 835,871,232,077,058 	 816,280,500,416.6 	 408,140,250,720.3 	 408,140,249,696.3 
12	 835,871,232,077,058 	 408,140,250,720.3 	 204,070,126,384.1 	 204,070,124,336.1 
13	 835,871,232,077,058 	 204,070,126,384.1 	 102,035,065,240.1 	 102,035,061,144.1 
14	 835,871,232,077,058 	 102,035,065,240.1 	 51,017,536,716.0 	 51,017,528,524.0 
15	 835,871,232,077,058 	 51,017,536,716.0 	 25,508,776,550.0 	 25,508,760,166.0 
16	 835,871,232,077,058 	 25,508,776,550.0 	 12,754,404,659.0 	 12,754,371,891.0 
17	 835,871,232,077,058 	 12,754,404,659.0 	 6,377,235,097.4 	 6,377,169,561.6 
18	 835,871,232,077,058 	 6,377,235,097.4 	 3,188,683,084.3 	 3,188,552,013.2 
19	 835,871,232,077,058 	 3,188,683,084.3 	 1,594,472,610.5 	 1,594,210,473.7 
20	 835,871,232,077,058 	 1,594,472,610.5 	 797,498,420.5 	 796,974,190.0 
21	 835,871,232,077,058 	 797,498,420.5 	 399,273,268.5 	 398,225,152.0 
22	 835,871,232,077,058 	 399,273,268.5 	 200,683,375.0 	 198,589,893.5 
23	 835,871,232,077,058 	 200,683,375.0 	 102,424,249.7 	 98,259,125.3 
24	 835,871,232,077,058 	 102,424,249.7 	 55,292,561.1 	 47,131,688.7 
25	 835,871,232,077,058 	 55,292,561.1 	 35,204,903.4 	 20,087,657.7 
26	 835,871,232,077,058 	 35,204,903.4 	 29,473,968.9 	 5,730,934.5 
27	 835,871,232,077,058 	 29,473,968.9 	 28,916,805.9 	 557,163.0 
28	 835,871,232,077,058 	 28,916,805.9 	 28,911,438.2 	 5,367.7 
29	 835,871,232,077,058 	 28,911,438.2 	 28,911,437.7 	 0.5 <<DONE in 29

I am very sorry but there are no errors in the test that are the same in many languages. Moreover, as you can see at the top of the page, 102 guys passed the Python kata. It is not because you don't find the same results as in the tests that there is an issue with the kata!-)