Question 1199748
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At the start, all 100 men are facing backwards.<br>
The men are then told to change their orientation each time their number is divisible by a number from 2 to 50.  So the number of times each man changes orientation is the number of positive divisors greater than 1 that his number has.<br>
An integer has an odd number of positive divisors if and only if it is a perfect square; all other integers have an even number of positive divisors.<br>
However, 1 is a positive divisor of every integer; and the directions for changing orientation only start with being divisible by 2.  So, before the final instruction (for all of the men in the second 50 to change orientation), the men whose numbers are perfect squares are the ones that are facing backwards.<br>
In the group of men on the left, numbered 1 to 50, there are 7 men with numbers that are perfect squares: 1, 4, 9, 16, 25, 36, and 49.  So among the first 50 men, 43 finish facing forwards and 7 finish facing backwards.<br>
Since the men that go on the mission are the ones who finish facing backwards, only 7 men from the first 50 go on the mission.<br>
In the group of men on the right, numbered 51 to 100, there are 3 with numbers that are perfect squares: 64, 81, and 100.  So among the second group of 50 men, 47 end up facing forwards and 3 end up facing backwards.  But then every one of those 50 men is instructed to change orientation one more time, leaving only 3 facing forwards and 47 facing backwards.<br>
ANSWER: The number of men facing backwards at the end, and therefore going on the mission, was 7 + 47 = 54.<br>