No known species of reindeer can fly. But there are 300,000 species of living organisms yet to be classified, and while most of these are insects and germs, this does not completely rule out flyimg reindeer that only Santa has ever seen.

There are over 2 billion children in the world. At an average (census) rate of 3.5 children per household, that's 91.8 million homes. One presumes there's at least one good child in each.

Santa has 31 hours of Christmas to work with, thanks to the different time zones and the rotation of the earth, assuming he travels east to west.

This works our to 822.6 visits per second. This is to say that for each household with good children, Santa has about 1/1000th of a second to park, hop out of the sleigh, jump down the chimney, fill up the stockings, distribute the remaining presents under the tree, eat whatever snacks have been left, get back up the chimney, get back into the sleigh and move on to the next house. Assuming that each of these 91.8 million stops are evenly distributed around the earth(which they are not) we are talking about 1.25km per household, a total trip of 115 million km, not counting stops to do what most of us must do at least once every 31 hours, plus feeding etc.

This means that Santa's sleigh is moving at 1125km/s, 3000 times the speed of sound. For purposes of comparison the fastest man-made vehicle on earth, the Ulysses space probe, moves at a pokey 43.8km/s. A conventional reindeer can run tops, 25km/h.

The payload on the sleigh adds another interesting element. Assumuing that each child gets nothing more than a medium sized Lego set (1kg), the sleigh is carrying 312,300 tonnes, not counting Santa, who is invariably described as being overweight. On land conventional reindeer can pull no more than 180kg. Even granting that "flying reindeer" could pull 10 times the normal amount, we cannot do the job with eight, or even nine. We need 178,500 reindeer. This increases the payload--not even counting the weight of the sleigh--to 353,430 tonnes. Again for comparison, this 4 times the weight of the liner Queen Elizabeth 2.

353,430 tonnes travelling at 1125km/s creates enormous air resistance-- this will heat up the reindeer in the same fashion as a spacecraft re-entering the earth's atmosphere. The lead pair of reindeer absorb 14.3 quintillion joules of energy. Per second. Each. In short they will burst into flame instantaneously, exposing the reindeer behind them, and creating deafening sonic booms in their wake. The entire reindeer team will be vaporised within 0.00426 seconds. Santa meanwhile, will be subjected to forces 17,500 times greater than gravity. A 115kg Santa would be pinned to the back of his sleigh by 2,012,500 kg of force.

In conclusion:-If Santa ever did deliver presents on Christmas Eve, he's dead now!!!