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Cutting cubes

I have before me a number of solid cubes. I make a single straight cut through each of them, avoiding cutting through any of the vertices. The resulting solids have between them the same number of even-sided faces as odd-sided faces. The number of cubes I started with is the minimum compatible with the information given above.

How many cubes did I start with?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 15 May. The Editor’s decision is final. Please send entries to Enigma 1745, New Å®ÉúСÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1739 Dodecagarden: The larger triangle was 147 times the area of the smaller

The winner John Crook of Betio, Republic of Kiribati