Question

Start with a list of numbers 1,2,....,2n. Remove two numbers a,b and add |a-b| to the list. Duplicates are allowed. For example, if n=2, you start with [1,2,3,4], you can remove 1 and 4 from the list and add |4-1|=3 to the list, leaving [2,3,3].Continue until one element is remaining. Prove or disprove that if the number remaining is even, n must be even.

Answer 1

for n = 1 , numbers will be [1,2] , it contains 2 elements. For this after removing the first and last number of the list, The remaining list will be [|2-1|] =[1] . It contain only one element, so can't reduce further.

for n = 2, numbers will be [1,2,3,4] . It will be converted to [2,3,3] -> [3,1] ->[2]

for n=3, numbers will be [1,2,3,4,5,6] . It will be converted to [2,3,4,5,5] ->[3,4,5,3]->[4,5,0]->[5,4]->[1]

for n=4, numbers will be [1,2,3,4,5,6,7,8] . It will be converted to [2,3,4,5,6,7,7]->[3,4,5,6,7,5]->[4,5,6,7,2]->[5,6,7,2]-> [6,7,3] ->[7,3] -> [4]

It is observed for n= 1 remaining no is [1] which is ODD
for n= 2 remaining no is [2] which is EVEN
for n= 3 remaining no is [1] which is ODD
for n= 4 remaining no is [4] which is EVEN  

Hence it is proved from the observation that that if the number remaining is even, n must be even and that if the number remaining is odd, n must be odd.