Question

I’m trying to write formulae that will split a given number into the sum of 4 other numbers.

The other numbers are 100,150,170 and 200 so the formula would be
x = a*100+b*150+c*170+d*200 where x is the given number and a,b,c,d are integers.

My spreadsheet is set up as where col B are x values, and C,D,E,F are a,b,c,d respectively (see below).

   B  |  C  |  D  |  E  |  F  |
  100    1     0     0     0
  150    0     1     0     0   
  200    0     0     0     1  
  250    1     1     0     0  
  370    0     0     1     1   
  400    0     0     0     2 

I need formulae for columns C,D,E,F (which are a,b,c,d in the formula)

Your help is greatly appreciated.

Answer 1

UPDATE:

Based on the research below, for input numbers greater than 730 and/or for all actually divisible input numbers use the following formulas:

100s: =CHOOSE(MOD(ROUNDUP([@number]/10;0); 20)+1;
       0;1;1;0;1;1;0;1;0;0;1;0;0;1;0;0;1;0;1;1)
150s: =CHOOSE(MOD(ROUNDUP([@number]/10;0); 10)+1;
       0;0;1;1;0;1;1;0;0;1)
170s: =CHOOSE(MOD(ROUNDUP([@number]/10;0); 5)+1;
       0;3;1;4;2)
200s: =CEILING(([@number]-930)/200;1) + 
       CHOOSE(MOD(ROUNDUP([@number]/10;0); 20)+1;
       4;1;2;0;2;3;1;3;1;2;4;2;3;0;2;3;0;3;0;1)

• MOD(x; 20) will return numbers 0 – 19, CHOOSE(x;a;b;...) will return n-th argument based on the first argument (1=>second argument, …)
• see more info about CHOOSE
• use , instead of ; based on your Windows language&region settings

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let’s start with the assumption that you want to preferably use 200s over 170s over 150s over 100s – i.e. 300=200+100 instead of 300=2*150 and follow the logical conclusions:

1. the result set can only contain at most 1 100, at most 1 150, at most 4 170s and unlimited number of 200s (i started with 9 170s because 1700=8×200+100, but in reality there were at most 4)

2. there are only 20 possible subsets of (100s, 150s, 170s) – 2*2*5 options

3. 930 is the largest input number without any 200s in the result set

4. based on observation of the data points, the subset repeats periodically for
   
   number = 740*k + 10*l, k>1, l>0 – i’m not an expert on reverse-guessing on periodic functions from data, but here is my work in progress (charted data points are from the table at the bottom of this answer)

• the functions are probably more complicated, if i manage to get them right, i’ll update the answer

• anyway for numbers smaller than 740, more tweaking of the formulas or a lookup table are needed (e.g. there is no way to get 730, so the result should be the same as for 740)

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Here is my solution based on lookup formulas:

Following is the python script i used to generate the data points, formulas from the picture and the 60-row table itself in csv format (sorted as needed by the match function):

headers = ("100s", "150s", "170s", "200s")
table = {}
for c200 in range(30, -1, -1):
    for c170 in range(9, -1, -1):
        for c150 in range(1, -1, -1):
            for c100 in range(1, -1, -1):
                nr = 200*c200 + 170*c170 + 150*c150 + 100*c100
                if nr not in table and nr <= 6000:
                    table[nr] = (c100, c150, c170, c200)
print("number\t" + "\t".join(headers))
for r in sorted(table):
    c100, c150, c170, c200 = table[r]
    print("{:6}\t{:2}\t{:2}\t{:2}\t{:2}".format(r, c100, c150, c170, c200))
__________

=IF(E$1<740; 0; INT((E$1-740)/200))
=E$1 - E$2*200
=MATCH(E$3; table[number]; -1)
=INDEX(table[number]; E$4)
=INDEX(table[100s]; E$4)
=INDEX(table[150s]; E$4)
=INDEX(table[170s]; E$4)
=INDEX(table[200s]; E$4) + E$2
__________

number,100s,150s,170s,200s
940,0,0,2,3
930,1,1,4,0
920,0,1,1,3
910,0,0,3,2
900,1,0,0,4
890,0,1,2,2
880,0,0,4,1
870,1,0,1,3
860,0,1,3,1
850,1,1,0,3
840,1,0,2,2
830,0,1,4,0
820,1,1,1,2
810,1,0,3,1
800,0,0,0,4
790,1,1,2,1
780,1,0,4,0
770,0,0,1,3
760,1,1,3,0
750,0,1,0,3
740,0,0,2,2
720,0,1,1,2
710,0,0,3,1
700,1,0,0,3
690,0,1,2,1
680,0,0,4,0
670,1,0,1,2
660,0,1,3,0
650,1,1,0,2
640,1,0,2,1
620,1,1,1,1
610,1,0,3,0
600,0,0,0,3
590,1,1,2,0
570,0,0,1,2
550,0,1,0,2
540,0,0,2,1
520,0,1,1,1
510,0,0,3,0
500,1,0,0,2
490,0,1,2,0
470,1,0,1,1
450,1,1,0,1
440,1,0,2,0
420,1,1,1,0
400,0,0,0,2
370,0,0,1,1
350,0,1,0,1
340,0,0,2,0
320,0,1,1,0
300,1,0,0,1
270,1,0,1,0
250,1,1,0,0
200,0,0,0,1
170,0,0,1,0
150,0,1,0,0
100,1,0,0,0
0,0,0,0,0