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Findings
None of the powers of two can be made. Also, a number can always be made if it is divisible by an odd number e.g. 20/5 = 4. The next step is to take the divisor, and minus one of it e.g. 5-1=4 Then half the answer e.g. 4/2 = 2 now, to form the sum of consecutives, take the answer of the first division - in this case, 4, and then you use the answer of the sum that got you 2, and use those to add that number of numbers inclusive above and below your first answer; 2+3+4+5+6=20. It's simpler than it sounds, but quite hard to explain. What this means is that all the numbers other than the powers of can be made (because all odd numbers can be made.
Proof
To understand it you have to understand that all odd numbers are themselves take one, halved, then you add one to one of halves (3= 5-1 (4) halved (2) then 2+(2+1) = 5. You can't do this with even numbers because even+odd=odd. As a work around, odd+odd=even, and so odd+odd+even=even. You split it into three sub-assemblies: 2 odds and one even. The even comes from dividing the number by an odd, then you get the odds from just less and more than the even equally, so they cancel each other out, leaving you effectively with the correct number of the first numbers to make your number. With twenty, 20/5=4, then you get 2+3+4+5+6 which make 2+6 (8), 3+5 (8) and 4 (4), effectively leaving you with 4+4+4+4+4=20 OR 4*5=20.
Further Questions
Is the rule that the powers of two can never be made always true? Can all numbers except the powers of two be made?