The Cargados Manufacturing Company fabricate what appear to be conventional six-sided dice, but each die is adulterated by the deliberate introduction of a spherical air bubble that occupies ²⁵⁄₁₂ π % of the volume. The dice have to look normal, so the air bubble must remain entirely within the die, but the surface of the bubble can meet the die surface.

After throwing the die, the probability of a given face being on top is directly proportional to the shortest distance from that face to the centre of the bubble. The bubble is situated so that the expected value of a single roll is minimised. If two such die are rolled simultaneously what is the expected value of their sum?

Please enter your answer as a decimal.

Mathsbombe Competition 2016 is organised by the The Department of Mathematics at The University of Manchester.
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