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Squares on cubes

Susan Denham

I HAVE three wooden cubes. On each face of each cube one of the digits from
0 to 8 is written, with each of those digits on more than one of the cubes.
The sum of the digits on each cube is the same.

By moving the cubes around I can make the top faces form many numbers (with
the 6 being also used for a 9);

for example,

In this way I can make all the three figure perfect squares.

How many of the three-figure perfect cubes cannot be made in this way?

A £10 book token will be awarded to the sender of the first correct
answer opened on Thursday 24 November. The Editor’s decision is final. Please
send entries to Enigma 794, Â鶹´«Ã½, King’s Reach Tower, Stamford
Street, London SE99 0BB. The winner of Enigma 789 was A. Brooks Harris of
Oxford.

Answer to Enigma 789

Fourth to first

(a) Borough v. Rangers; Rangers v. United; (b) City v. United.