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 Three Cubed (Posted on 2005-11-19)
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.

 Subject Author Date your solution Morgan 2005-11-30 14:12:27 re: three cubed melissa 2005-11-20 17:32:14 three cubed melissa 2005-11-20 17:28:56 re(2): my solution - before looking at the goFish pcbouhid 2005-11-19 18:03:48 Solution Vernon Lewis 2005-11-19 09:12:10 re: my solution - before looking at the goFish Vernon Lewis 2005-11-19 08:53:50 my solution - before looking at the goFish pcbouhid 2005-11-19 07:47:34 Solution goFish 2005-11-19 06:43:44

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