Finding the possible location of points The numbers 1,2,....6 are to be placed in some order at the points A,B,.....F in the figure below. How many ways can the numbers be placed so that each sum of consecutive pairs of points is odd?
 A: HINT: The sum of two integers is odd if and only if one of the integers is odd and the other is even; this greatly limits the possible sets of positions for the even integers. How many ways are there to choose the points that will be assigned even integers? Once you’ve chosen those points, how many ways are there to assign even integers to them? How many ways are there then to assign the odd integers>
A: You need to alternate an odd and an even number to obtain an odd sum. We start at A, we can put an odd number (3 choices), then an even number on B (3 choices), then an odd number on C (2 remaining choices), etc., that is $3*3*2*2*1*1$ possibilities. Now, we could have started with an even number at A and repeated the process, so we just need to double the above result. Hence, the answer is 72. 
A: As stated, this is a math problem on Brilliant and solutions are available there. It gets reposted here because OP is merely interested in getting the correct numerical answer.
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