Suppose that |a-b|+|b-c|+|c-d|+.......+|m-n|+|n-o|+............|x-y|+|y-z|+|z-a|=20. What is the maximum possible value of |a-n|?
Because it is circularly symmetrical, all 26 variables cannot have the same value (otherwise the sum would be zero).
To maximize the value of |a-n|, we need
|a-n|=10 which in the integer domain gives
±(0,-10),±(1,-9),±(2,-8),...±(5,-5),
add: ...and the remaining 24 variables equal to zero.
To find the maximum possible value of |a-n|, we need to understand the given equation and determine the conditions that would maximize this value.
The given equation |a-b| + |b-c| + |c-d| + ... + |m-n| + |n-o| + ... + |x-y| + |y-z| + |z-a| = 20 implies that the sum of the absolute differences between consecutive variables counts up to 20.
Let's consider the variables in consecutive order: a, b, c, ..., m, n, o, ..., x, y, z. Since the sum of the absolute differences between consecutive variables is limited to 20, we can conclude that each individual absolute difference term cannot exceed 20.
To maximize |a-n|, we need to ensure that the absolute difference between a and n is as high as possible. Since the given equation does not provide any other constraints, we can consider a circular arrangement for the variables, where z comes directly before a.
In this arrangement, the equation can be rewritten as:
|a-z| + |z-b| + |b-c| + |c-d| + ... + |m-n| + |n-o| + ... + |x-y| + |y-z| = 20
Now, to maximize |a-n|, we need to minimize the absolute difference between a and z while ensuring that the sum of absolute differences between all the other variables remain at 20.
To minimize |a-z|, we can set a = z. This configuration allows the maximum possible value for |a-n| since it reduces |a-n| to |a-a| = 0.
Therefore, the maximum possible value of |a-n| is 0.
To find the maximum possible value of |a-n|, we need to understand the relationship between the given equation and the value of |a-n|.
Let's break down the given equation first. We have:
|a - b| + |b - c| + |c - d| + ... + |m - n| + |n - o| + ... + |x - y| + |y - z| + |z - a| = 20
Since the absolute value of a difference always yields a non-negative value, we can conclude that the sum of these absolute values must be equal to or greater than 20.
Now, consider the expression |a - n|. We want to maximize this value.
We can see that |a - n| is a single term in the given equation. In order to maximize |a - n|, we want its term to be as large as possible.
To achieve this, we want to choose values for a and n that are as far apart as possible.
One way to do this is to consider a linear arrangement of the alphabet, where a, b, c, d, ... , x, y, and z are consecutive letters. In this case, a and n would be the endpoints of the line.
Assuming that z is chosen as the initial point (to make the expression easier), n would be the letter preceding z, and a would be the letter following z, which wraps around to the beginning.
With this arrangement, |a - n| would be equal to |a - z| since a - n accounts for the entire alphabet loop.
Considering the given equation, we have:
|a - b| + |b - c| + ... + |x - y| + |y - z| + |z - a| = 20
Since all other terms except |a - z| have a non-zero value, we want to minimize their contribution while still satisfying the equation. This can be done by making each term equal to 1.
Hence, we can rewrite the equation as:
|a - z| + (1 + 1 + ... + 1) = 20
Now, it is clear that the sum of the 25 terms with a value of 1 is 25.
Simplifying further:
|a - z| + 25 = 20
|a - z| = 20 - 25
|a - z| = -5
Since the absolute value of a real number is always non-negative, there are no real solutions for |a - z| = -5. Thus, the maximum possible value of |a - n| does not exist.
In conclusion, there is no maximum possible value for |a - n| given the equation.