:|~ Maths Problem (RESOLVED)

[da_silvy]

Bob writes down 6 consecutive positive integers and splits them up into two groups. For each group, he finds the sum and then finds the difference of these two new numbers. What is the smallest non-negative number he can get?

i think the answer is 1, it may be 0

But who can help me prove it?

[Ambivalentiowa]

ANy 6 consecutive numbers when split into two groups,
one group will be even and the other will be odd.

Therefore, there will never be a difference of 0.

[da_silvy]

thanks very much

just what i wanted

[BigGlass]

1. I believe in Ambivalentiowa
2. Conjecture: The less integer for the problem is greater than 1
3. The conjecture is false, we will prove by counter example

Six consecutive number are x, x+1, x+2, x+3, x+4, x+5 or x+1, x+2, x+3, x+4, x+5, x+6. I’ll use the last for convenience.
We can change the number 1, 2, 3, 4, 5 and 6 variables and split in two groups, so,
x + a0 + x + a1 + x + a2 – (x + a3 + x + a4 + x + a5) where 0 < ai <= 6 and ai <> aj
3x + (a0 + a1 + a2) – 3x – ( a3 + a4 + a5)
(a0 + a1 + a2) – ( a3 + a4 + a5)
now we have only 20 combinations (6!/(6-3)!3!) “Brute force” is feasible, then
(6+5+4) - (3+2+1) = 9
(6+5+3) - (4+2+1) = 7
(6+5+2) - (4+3+1) = 5
(6+5+1) - (4+3+2) = 3
(6+4+3) - (5+2+1) = 5
(6+4+2) - (5+3+1) = 3
(6+4+1) - (5+3+2) = 1 (Bingo!)

[BigGlass]

1. I believe in Ambivalentiowa
2. Conjecture: The less integer for the problem is greater than 1
3. The conjecture is false, we will prove by counter example

Six consecutive number are x, x+1, x+2, x+3, x+4, x+5 or x+1, x+2, x+3, x+4, x+5, x+6. I’ll use the last for convenience.
We can change the number 1, 2, 3, 4, 5 and 6 variables and split in two groups, so,
x + a0 + x + a1 + x + a2 – (x + a3 + x + a4 + x + a5) where 0 < ai <= 6 and ai <> aj
3x + (a0 + a1 + a2) – 3x – ( a3 + a4 + a5)
(a0 + a1 + a2) – ( a3 + a4 + a5)
now we have only 20 combinations (6!/(6-3)!3!) “Brute force” is feasible, then
(6+5+4) - (3+2+1) = 9
(6+5+3) - (4+2+1) = 7
(6+5+2) - (4+3+1) = 5
(6+5+1) - (4+3+2) = 3
(6+4+3) - (5+2+1) = 5
(6+4+2) - (5+3+1) = 3
(6+4+1) - (5+3+2) = 1 (Bingo!)

[Ambivalentiowa]

Thanks for explaining it BigGlass, i forgot to put an example.

[da_silvy]

Thanks for that

I knew it was 1