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Algebra Level 4

The fastest supercomputer Sunway TaihuLight is commissioned to generate all $4 \times 4$ pixel, 8 bit grayscale images.

The supercomputer can generate $10^{15}$ images per second. Assume that one day has exactly 86400 seconds, and a year has $365.25$ days (on average). Now for each century (hundred years) that passes without the task getting complete, a single grain of rice (weighing $\dfrac1{64}$ grams) is added to a pile of rice grains.

When the task is complete, the heap weigh $T$ kilograms.

Find the value of $\lfloor T \rfloor$ ?

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 1684827738.

1 solution

Janardhanan Sivaramakrishnan
Jul 30, 2016

There are $16 = 4 \times 4$ pixels in each image and each of these pixels could take $2^8 = 256$ values in an 8-bit grayscale image.

Therefore, there are a total of $256^{16}$ images to be generated.

Now, with the specifications given, the number of kilograms of grain would come out to be

$T =\frac{256^{16}}{10^{15}\times 86400 \times 365.25 \times 100} \times \frac{1}{64} \times \frac{1}{1000} = 1.6848277382119247001087624664 ...× 10^9$

Which gives $\lfloor T \rfloor = \boxed{1684827738}$

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The answer is 1684827738.

1 solution

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