Dividing an even number into N parts each part being a multiple of 2

Question

Let's assume I have the number 100 which I need to divide into N parts each of which shouldn't exceed 30 initially. So the initial grouping would be (30,30,30). The remainder (which is 10) is to be distributed among these three groups by adding 2 to each group in succession, thus ensuring that each group is a multiple of 2. The desired output should therefore look like (34,34,32).

Note: The original number is always even.

I tried solving this in Python and this is what I came up with. Clearly it's not working in the way I thought it would. It distributes the remainder by adding 1 (and not 2, as desired) iteratively to each group.

num = 100
parts = num//30  #Number of parts into which 'num' is to be divided

def split(a, b):
  result = ([a//b + 1] * (a%b) + [a//b] * (b - a%b))
  return(result)

print(split(num, parts))

Output:

[34, 33, 33]

Desired output:

[34, 34, 32]

Answer 1

Simplified problem: forget about multiples of 2

First, let's simplify your problem for a second. Forget about the multiples of 2. Imagine you want to split a non-necessarily-even number n into k non-necessarily-even parts.

Obviously the most balanced solution is to have some parts be n // k, and some parts be n // k + 1.

How many of which? Let's call r the number of parts with n // k + 1. Then there are k - r parts with n // k, and all the parts sum up to:

   (n // k) * (k - r) + (n // k + 1) * r
== (n // k) * (k - r) + (n // k) * r + r
== (n // k) * (k - r + r) + r
== (n // k) * k + r

But the parts should sum up to n, so we need to find r such that:

n == (n // k) * k + r

Happily, you might recognise Euclidean division here, with n // k being the quotient and r being the remainder.

This gives us our split function:

def split(n, k):
    d,r = divmod(n, k)
    return [d+1]*r + [d]*(k-r)

Testing:

print( split(50, 3) )
# [17, 17, 16]

Splitting into multiples of 2

Now back to your split_even problem. Now that we have the generic function split, a simple way to solve split_even is to use split:

def split_even(n, k):
    return [2 * x for x in split(n // 2, k)]

Testing:

print( split_even(100, 3) )
# [34, 34, 32]

Generalisation: multiples of m

It's trivial to do the same thing with multiples of a number m other than 2:

def split_multiples(n, k, m=2):
    return [m * x for x in split(n // m, k)]

Testing:

print( split_multiples(102, 4, 3) )
# [27, 27, 24, 24]

Answer 2

This solution is not very clear and easy to follow but it does not need any loops.

Full code:

def split(a,b):
     lower = (a//b//2) * 2
     num = a % (b*2) // 2
     return [lower + 2] * num + [lower] * (b - num)

Explanation:

• First get the value of all parts: We round the result of the division (value // parts) down to the next even value ((x // 2) * 2)
• To get the number of higher values: We use the remainder of the division of a in double as many parts and divide it by two to compensate the multiplication
• last: higher numbers are just lower + 2 times the computed number of higher values and lower numbers are filling the other spaces