Even some asks in sincere, the question of prayer and fasting times at the North and South poles are asked by many to find mistakes in Islam due to the lack of technical information. In Quran, Allah describes the times directing the Sun movements to clarify the times of prayers and fasting.
The statement of “Six months day- six months night” is not actually correct. The correct one is “six months of daylight, six months of dark”. Furthermore, because of the misunderstanding of the situation at the poles, some people think that Sun has no movement and it stands still during 6 months at poles. And they asked “How can we determine the fasting and prayer times at Poles?” This one is absolutely wrong. In the Picture below, you can see the motion of the Sun. At the Poles, the sun always moves during the day; and morning and night periods can be understood easily. The nearest position of  the Sun to the horizon is accepted as night, the top position is accepted as midday.

During the six months of dark, when the Sun is below the horizon, the motion of Sun also can be determined clearly. Prayer and fasting times can be adjusted easily according to that. Today, thousands of muslims in the region are fasting and praying according to those motions.

But, the most important subject that people have ignored so far is Allah wants us to be sincere and piety/taqwa in Quran. For instance, in Quran, Allah orders us about Eid al Adha/sacrifice of animals;

“Their meat will not reach Allah , nor will their blood, but what reaches Him is piety from you…” Quran 22/37
It means Allah is not interested in the material parts of our devotions, only piety from us is valuable for him.
“They who are during their prayer humbly submissive.” Quran 23/2
It is useless to search for mistakes and failure in Quran. There is no failure or contradiction in Quran. Quran is absolutely clear, fully detailed, fullfilled in truth and in justince and perfect.

You can take the Ramadan and prayers timetable for the whole World from the the following link…

http://imsakiye.suleymaniyetakvimi.com/