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Two expressions (Posted on 2011-12-22) Difficulty: 2 of 5
N is the smallest number which can be represented by two different sums of 4 positive (not necessarily distinct) cubes.
Find N and the corresponding sums.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution (spoiler) Comment 1 of 1

DEFDBL A-Z
OPEN "twoexps.txt" FOR OUTPUT AS #2
FOR a = 1 TO 40
 a3 = a * a * a
FOR b = a TO 40
 b3 = b * b * b
FOR c = b TO 40
 c3 = c * c * c
FOR d = c TO 40
 d3 = d * d * d
 sum = a3 + b3 + c3 + d3
 PRINT #2, USING "#######"; a3; b3; c3; d3; sum
NEXT
NEXT
NEXT
NEXT
CLOSE

The output file is then sorted on the sum column and read by:

OPEN "twoexps.txt" FOR INPUT AS #1
CLS

DO
  prev$ = l$
  LINE INPUT #1, l$
  IF MID$(l$, 30) = MID$(prev$, 30) THEN
     PRINT prev$
     PRINT l$
     PRINT
     ct = ct + 1: IF ct > 12 THEN END
  END IF

LOOP UNTIL EOF(1)

the result is:

     1      1      1    216    219
    27     64     64     64    219
     1      1    125    125    252
     1      8     27    216    252
     1      8    125    125    259
     8      8     27    216    259
     8     27     27    216    278
     1     27    125    125    278
     1     64    125    125    315
     8     27     64    216    315
     8     27    125    216    376
     1    125    125    125    376
     8     27    216    216    467
     1    125    125    216    467
    27     27    125    343    522
     1      1      8    512    522
     1    125    125    343    594
     8     27    216    343    594
     1     64    125    512    702
     8      8    343    343    702
     1      1     27    729    758
     8     64    343    343    758
     8     27    216    512    763
     1    125    125    512    763
    64     64    125    512    765
     1      8     27    729    765
    


so the answer is 219 = 1+1+1+216 = 27+64+64+64.    

BTW, eyeballing the output, allowing it to continue past the above, indicates the first instance that would satisfy disallowing repeated cubes within a given sum would result in the following as the lowest:


  1  +   8  +  27 + 1000 = 1036
 27  +  64  + 216 +  729 = 1036
 
 
If you want no repeats whatsoever among the eight cubes, then:

   1  + 125 +  512 + 1000 = 1638
  27  +  64 +  216 + 1331 = 1638


  Posted by Charlie on 2011-12-22 14:09:17
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