Minimum number of steps to reach a given number

I need to calculate the minimum number of ways to reach a value, x, from value n, by adding/subtracting a list of values, l, to n.

For example: Value n = 100, value X = 45 List, l,: 50,6,1

The best way to do this is to say: 100-50-6+1 = 45

I want a programme to work this out for any value of x and n given list, l

I am really struggling to outline how I would write this.

I am confused about how to overcome the following issues:

How to inform the programme if I should attempt an addition or subtraction and how many times this should be done. For example I might need to subtract, then add, then subtract again to reach a solution
How do I include enough for/while loops to ensure I can provide a solution for all possible input values

Has anyone come across an issue like this before and have any ideas how I could outline the code for such a solution (I am using Python if it helps direct me towards learning about particular functions available that could assist me)

Thanks

This is my attempt so far but I am stuck

inputA = ""
while inputA == "":
    inputA = input("""Please enter two numbers, separated by a comma.
                The first value should indicate the number of jugs:

                The second value should indicate the volume to be measured

                """)

itemList = list(inputA.split(","))
valueToMeasure = int(itemList[1])

inputB = ""

while inputB == "":
    inputB = input("Plese enter the volumes for the {} jug(s) listed: ".format((itemList[0])))

    if len(inputB.split(",")) != int(itemList[0]):
        inputB = ""

TargetVolume = itemList[1]
jugSizes = inputB.split(",")

print("Calculating: smallest number of steps to get", TargetVolume, "ml using jugs of sizes:", jugSizes)


jugSizes.sort()
jugSizes.reverse()
largestJug = int(jugSizes[0])

ratioTable = {}
for item in jugSizes:
    firstVal = int(jugSizes[0])

    itemV = int(item)
    valueToAssign = firstVal/itemV

    ratioTable[int(item)] = int(valueToAssign)

taskPossible = True

if valueToMeasure > largestJug:
    print ("Impossible task")
    taskPossible = False

newList = jugSizes
if taskPossible == True:
    for item in jugSizes:
        if item < TargetVolume: break
        newList = newList[1:]
        newDict  = {}
        for itemA in ratioTable:
            if int(itemA) < int(item):
                newDict[itemA]= ratioTable[itemA]
        print ("Do work with these numbers:", newDict)

Solution

This is how I would approach the problem if I understand correctly.

X = 45
largest_jug = measured = 100
jug_sizes = [50, 6, 1]
steps = []
jug_to_use = 0

while measured != X:
    if jug_to_use < len(jug_sizes) - 1: # we have smaller jugs in reserve
        error_with_large_jug = min([abs(measured - jug_sizes[jug_to_use] - X), abs(measured + jug_sizes[jug_to_use] - X)])
        error_with_small_jug = min([abs(measured - jug_sizes[jug_to_use + 1] - X), abs(measured + jug_sizes[jug_to_use + 1] - X)])
        if error_with_small_jug < error_with_large_jug:
            jug_to_use += 1
    if measured > X:
        measured -= jug_sizes[jug_to_use]
        steps.append(('-', jug_sizes[jug_to_use]))
    else:
        measured += jug_sizes[jug_to_use]
        steps.append(('+', jug_sizes[jug_to_use]))
print(steps)

Yielding

[('-', 50), ('-', 6), ('+', 1)]

It basically starts by using the largest jug, until it's in range of the next size and so on. We can test it with randomly sized jugs of [30, 7, 1] and see it again results in an accurate answer of [('-', 30), ('-', 30), ('+', 7), ('-', 1), ('-', 1)].

Important notes:

jug_sizes should be ordered largest to smallest
This solution assumes the X can be reached with the numbers provided in jug_sizes (otherwise it will infinitely loop)
This doesn't take into account that a jug size can make the target unreachable (i.e. [50, 12, 5] where the 12 size should be skipped, otherwise the solution is unreachable
This assumes every jug should be used (related to above point)

I'm sure you could figure out solutions for all these problems based on your specific circumstances though