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:

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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?

There are 5 3 digit cubes - 125; 216; 343; 512 and 729

So the middle digit of the 5 digit number can only be 2; 3; 5; 6 or 9.

Of the 14 cubes that meet this requirement only 3 meet the further requirement of 2 four digit cubes (and 2 other digits) using 9 different digits. Two of these - 17576 and 50653 are missing the digit 8. So solution is 68921; 4096; 1331; 729 (missing 5)