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Three Cubed (Posted on 2005-11-19) Difficulty: 3 of 5
Harry, Tom and I each replaced each asterisk in the diagram below with a single digit, such that the two horizontal 4-digit numbers, the horizontal 3-digit number and the vertical 5-digit number were all perfect cubes:

****
   *
 ***
   *
****

We each found a different solution that used exactly nine of the ten digits 0-9. Harry's unused digit was the same as Tom's.

What was my solution?

  Submitted by Charlie    
Rating: 3.0000 (4 votes)
Solution: (Hide)
The 3-digit cubes are:
125
216
343
512
729

The 4-digit cubes are:
1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000 9261 The 5-digit cubes are:
10648 12167 13824 15625 17576 19683 21952 24389 27000 29791 32768 35937 39304 42875 46656 50653 54872 59319 64000 68921 74088 79507 85184 91125 97336

The following program tries all the ways of fitting these into the diagram, and checks if each solution uses all but one of the ten possible digits:

DEFDBL A-Z 
CLS 
DIM dig3$(5), dig4$(12), dig5$(25) 
FOR i = 1 TO 46 
  i3 = i * i * i 
  SELECT CASE i3 
    CASE 100 TO 999 
     no3s = no3s + 1 
     dig3$(no3s) = LTRIM$(STR$(i3)) 
    CASE 1000 TO 9999 
     no4s = no4s + 1 
     dig4$(no4s) = LTRIM$(STR$(i3)) 
    CASE 10000 TO 99999 
     no5s = no5s + 1 
     dig5$(no5s) = LTRIM$(STR$(i3)) 
  END SELECT 
NEXT 
FOR first4 = 1 TO no4s 
 tf4$ = dig4$(first4) 
 FOR the5 = 1 TO no5s 
  t5$ = dig5$(the5) 
  IF LEFT$(t5$, 1) = RIGHT$(tf4$, 1) THEN 
    FOR the3 = 1 TO no3s 
      t3$ = dig3$(the3) 
      IF MID$(t5$, 3, 1) = RIGHT$(t3$, 1) THEN 
       FOR last4 = 1 TO no4s 
        tl4$ = dig4$(last4) 
        IF RIGHT$(t5$, 1) = RIGHT$(tl4$, 1) THEN 
         s$ = "0123456789" 
         x$ = tf4$ + tl4$ + t3$ + t5$ 
         FOR i = 1 TO LEN(x$) 
           ix = INSTR(s$, MID$(x$, i, 1)) 
           IF ix THEN 
             s$ = LEFT$(s$, ix - 1) + MID$(s$, ix + 1) 
           END IF 
         NEXT i 
         IF LEN(s$) = 1 THEN 
          PRINT tf4$ 
          PRINT "   "; MID$(t5$, 2, 1) 
          PRINT " "; t3$ 
          PRINT "   "; MID$(t5$, 4, 1) 
          PRINT tl4$, s$ 
          PRINT 
          ct = ct + 1 
         END IF 
        END IF 
       NEXT last4 
      END IF 
    NEXT the3 
  END IF 
 NEXT 
NEXT 
PRINT ct 

It produces:

1331
   7
 125
   7
4096          8

3375
   0
 216
   5
4913          8

4096
   8
 729
   2
1331          5

The unused digit is shown to the right. The unique one is for the third solution, which was therefore the solution of the narrator.

Adapted from New Scientist's Enigma number 1353, published in the August 13-19, 2005 issue.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
your solutionMorgan2005-11-30 14:12:27
re: three cubedmelissa2005-11-20 17:32:14
three cubedmelissa2005-11-20 17:28:56
re(2): my solution - before looking at the goFishpcbouhid2005-11-19 18:03:48
SolutionSolutionVernon Lewis2005-11-19 09:12:10
Questionre: my solution - before looking at the goFishVernon Lewis2005-11-19 08:53:50
Solutionmy solution - before looking at the goFishpcbouhid2005-11-19 07:47:34
SolutionSolutiongoFish2005-11-19 06:43:44
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