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I was able to answer the question logically but could not account for the exact values of fraction of the total radiation, so please give some good way to exactly calculate the fraction of the total radiation as detected by detector. Please help.

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The source of the radiation is above O. Construct 3 more squares which are identical to $OBCD$ as shown in the diagram.

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Similarly, construct 6 big squares to form a cube. The source of radiation is at the center of the cube, right above O. So each face of the cube would get $\frac{1}{6}$ of the radiation.

Now go back to the first image. Each face consists of 4 identical squares. The radiation hitting each face is shared equally by each part of the face. Therefore, the square $OBCD$ gets $\frac{\frac{1}{6}}{4}$, i.e: $ \frac{1}{24}$ of the total radiation.

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    $\begingroup$ This is a really lovely answer! $\endgroup$ – Floris Feb 10 '17 at 23:11

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